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xtremlove
The sides of a right triangle have a relationship as depicted below, where x is the length of the lo...
4 months ago
Q:
The sides of a right triangle have a relationship as depicted below, where x is the length of the longer leg in inches.The area of this triangle, A, is equal to the perimeter of this triangle.Create a system of equations to model this situation. Determine if there are any solutions, and, if possible, whether or not they are viable.How many total possible solutions are there to this system? Of any possible solutions, how many are viable solutions for this situation?
Accepted Solution
A:
The equation to represent the area of the triangle would be: y = 1/2(x²) - (7/2)x
The equation to represent the perimeter of the triangle would be: y = 3x - 6
The solutions to the system would be (12, 30) or (1, -3). The only viable solution is (12, 30).
Explanation The area of a triangle is found using the formula A = 1/2bh | 677.169 | 1 |
A triangle has two corners with angles of # pi / 4 # and # pi / 4 #. If one side of the triangle has a length of #16 #, what is the largest possible area of the triangle?
1 Answer
Explanation:
The angles of a triangle have to sum up to #pi# and since two of the angles are both #pi/4#, the remaining angle must be #pi/2#. Knowing special triangles, this is a right triangle where the two legs are the same length and the hypotenuse is #sqrt2# times the legs.
One of the sides of the triangle has a length of #16#. We can set that to be either the leg length or the hypotenuse length. Intuitively setting it as the leg length will give a larger triangle because the sides will be longer: #16, 16, and 16sqrt2~~22.6# instead of #16/sqrt2~~11.3, 16/sqrt2~~11.3, and 16#.
Since our largest triangle has the largest area, we know that the first set is what we're looking for.
Since this is a right triangle with legs 16 and 16, the area is #16^2/2=128# | 677.169 | 1 |
Tuesday, March 25, 2014
Archimedes' Twin Circles
As with many visual theorems, my first impulse was to make a dynamic visualization. Off to GeoGebra!
But I quickly hit a snag... I didn't know how to construct the tangent circles. I made the basic set up and then starting monkeying around. Eventually I thought about how to just make a circle tangent to one of the interior circles and the line. I made a tracing point with the distance to the interior circle and the dividing line and traced along where it would be equidistant - a requirement for a circle tangent to both.
Of course - a parabola. To be tangent to a circle and a line is like the definition of a parabola as all the points equidistant from the directrix and focus. The point D had to be on the parabola, which helped me to find the directrix. The focus had to be the center of the circle.
So then the other parabolas weren't too hard to find. The center of one of the twins had to be both on a parabola of tangents to an interior circle and the line, and a parabola for the enclosing circle and the line.
And now we can see that Archimedes result was true in general.
This is a special case of the Apollonian Circle problem (finding a circle tangent to three non-concentric circles &/or lines), and I feel like it was helpful in deepening my understanding of that. To be specific, a special case of the Circle-Circle-Line special case. But it was fun.
Now that we can construct them, how would you prove the twin-ness of these circles?
2 comments:
Well! You have shared superb article, the above comment, I don't like it. I think we appreciated this job. if someone want to ask question then share with detail. Any i really like and appreciated your work | 677.169 | 1 |
Geometry Vocabulary (4th Grade)
Description: Students can use this deck to identify and describe points, lines, line segments, rays, and angles (including vertices). Some symbolic notation is included. They will also identify and describe parallel, perpendicular, and intersecting line segments in practical situations and in plane and solid figures.
Supports Virginia SOL 4.10. | 677.169 | 1 |
But sir can you just tell me how that perpendicular distance formula is used. I would be really grateful to you.
Thanx once again
Prajwal kr
49
Points
11 years agoJainam Shroff
26
Points
7 years ago
Please Provide The Derivation,Sir
In Reply ToManonmani
11
Points
7 years ago
why cant we use the method used by bhaveen kumar...can i please know what is the mistake in it and why arent the answers matching???
Teju
11
Points
7 years ago
Mannon Mani, we cannot use that nethod because he took the points in both the curve and line with respect to t, which is incorrect as they are non intersecting. They cannot be taken with the same parameter
Yasaschandra Dvs
25
Points
7 years ago
what if i use the distance formulae and ill get s= mag( t-t^2-1/ sqrt 1+1) => s= t^2-t+1/sqrt(2) { since t^2-t+1 = (t-1/2) ^2+3 /4 >0} and then ds/dt = 1/sqrt2 (2t-1) | 677.169 | 1 |
The perpendicular from A on side BC of a ΔABC intersects BC at D such that DB = 3CD .Prove that 2AB2=2AC2+BC2.
Open in App
Solution
Given that in ΔABC, we have AD⊥BCandBD=3CD
In right angle triangles ADB and ADC, we have AB2=AD2+BD2...(i) AC2=AD2+DC2...(ii) [By Pythagoras theorem]
Subtracting equation (ii) from equation (i), we get AB2−AC2=BD2−DC2 =9CD2−CD2[∴BD=3CD] =8CD2 =8(BC4)2[Since,BC=DB+CD=3CD+CD=4CD]
Therefore, AB2−AC2=BC22 ⇒2(AB2−AC2)=BC2 ⇒2AB2−2AC2=BC2 ∴2AB2=2AC2+BC2 | 677.169 | 1 |
(between ?OBJ1 ?OBJ2 ?OBJ3) means that ?OBJ2 is spatially located between ?OBJ1 and ?OBJ3. Note that this implies that ?OBJ2 is directly between ?OBJ1 and ?OBJ3, i.e. the projections of ?OBJ1 and ?OBJ3 overlap with ?OBJ2 | 677.169 | 1 |
In geometry books in French, there has traditionally been a very clear distinction between 'orthogonal' lines and 'perpendicular' lines in space. Two lines are called perpendicular if they meet at a right angle. Two lines are called orthogonal if they are parallel to lines that meet at a right angle. Thus orthogonal lines could be skew (i.e., they need not meet), whereas perpendicular lines always intersect. [Edit: Evidence shows that this distinction most likely arose around the turn of the twentieth century. See below for details.]
Looking around on Quora, Answers.com and here, I have found numerous assertions that, in English, there is no difference whatsoever between 'orthogonal' and 'perpendicular.' However, given the situation in French, I have a gut feeling that the same distinction must once have been observed in English as well, but since there is now a greater focus on vectors (for which the concepts coincide) than on lines, it has gradually been lost. I would like confirmation of this, if possible.
My question, then, is as follows:
How have the two concepts referred to above as 'orthogonal' and 'perpendicular' lines historically been denoted in English and other major languages?
The best answers will include references to authoritative sources.
Edit. Zyx has provided an answer referring to Rouché and Comberousse's geometry text from 1900, where the word perpendiculaire is used for what we have called orthogonal here. This strongly suggests that, contrary to what I had assumed, even French usage has not been unchanging over time.
So Zyx may be correct in questioning my premise, and I am beginning to suspect that even in France, the use of orthogonal in the sense discussed here may have been introduced in the twentieth century. Let me give an example taken from 1952 geometry text that illustrates this usage (Géométrie dans l'espace: Classes de Première C et Moderne, 1952, Dollon and Gilet):
Nearly identical conventions are found in Géométrie: Classe de Seconde C, 1964, by Hémery and Lebossé, except that they allow "orthogonal" lines to meet (thus perpendicular implies orthogonal, but not conversely):
And Géométrie Élémentaire, 1903, by Vacquant and Macé de Lépinay agrees with Hadamard.
My conclusion is that I was much too quick in my question to call the distinction "traditional." It is most likely to have appeared in France sometime in the early to mid-twentieth century. (To pinpoint the date better, it would be best to check what was done in textbooks in the 1925-1940 period, such as those of P. Chenevier and H. Commissaire, but I don't have access to these. Vectors evidently first appeared in French school curricula in 1905. However, the scalar product was not taught systematically until 1947, so that would seem a possible time for the expression "orthogonal lines" to have been introduced.)
The examples given by Zyx show that usage in English in fact mirrors the earlier French usage, i.e. "perpendicular" is used everywhere. And I presume that the terms "skew perpendicular" and "intersecting perpendicular" would only be used where an author felt the distinction was needed. (In many cases, it will be clear from context whether two lines meet.)
Edit. The "new" French terminology dates at least from the turn of the century. Here is an excerpt from Cours de Géométrie élémentaire: à l'usage des élèves de mathématiques élémentaires, de mathématiques spéciales; des candidats aux écoles du Gouvernement et des candidats à l'Agrégation (1899) by Niewenglowski and Gérard, which was intended for both high-school and university-level students. This book is in fact referred to by Lebesgue in his Leçons sur l'intégration.
Thus these authors, unlike Hadamard, Rouché and Vacquant, appear to have a preference for droites orthogonales when the lines are not coplanar. However, this was not a hard-and-fast rule, and they allow that perpendiculaires can also "sometimes" be used in this case. The distinction only seems to have become settled later on.
$\begingroup$In all texts I encountered so far, this distinction is not made. Usually it is mentioned whether two (orthogonal, i.e. having orthogonal vectors) lines intersect or not. I never read a textbook that explicitly mentiones the distinction by using different words for it. I guess the same use of terminology as in French would be appropriate.$\endgroup$
5 Answers
5
ORTHOGONAL is found in English in 1571 in A geometrical practise named Pantometria by Thomas Digges (1546?-1595): "Of straight lined angles there are three kindes, the Orthogonall, the Obtuse and the Acute Angle." (In Billingsley's 1570 translation of Euclid, an orthogon (spelled in Latin orthogonium or orthogonion) is a right triangle.) (OED2).
also
ORTHOGONAL VECTORS. The term perpendicular was used in the Gibbsian version of vector analysis. Thus E. B. Wilson, Vector Analysis (1901, p. 56) writes "the condition for the perpendicularity of two vectors neither of which vanishes is A·B = 0." When the analogy with functions was recognised the term "orthogonal" was adopted. It appears, e.g., in Courant and Hilbert's Methoden der Mathematischen Physik (1924).
There are also notes on orthogonal matrix and orthogonal function, and orthocenter (the last of which includes an anecdote about the coining of the term in 1865).
PERPENDICULAR was used in English by Chaucer about 1391 in A Treatise on the Astrolabe. The term is used as a geometry term in 1570 in Sir Henry Billingsley's translation of Euclid's Elements.
Note that Billingsley also appears in the "orthogonal" entry above. A deeper dive into his translation of Elements may be in order, to see if he explains his own thinking about the distinction between "perpendicular" and "ortho[gonal]".
Anecdotally, I (an American) was formally introduced to "orthogonal" in the context of vectors in Pre-Calculus. (The term may have been mentioned in passing when we learned about orthocenters in Geometry.) So, the term to me has always connoted a directional relationship independent of position. I've also seen the term "perpendicularly skew" for lines in space. Be that as it may ... I don't appear to be alone in using "orthogonal" and "perpendicular" interchangeably ---"perpendicular" just seems friendlier to use with students--- but in formal circumstances, I would probably be inclined to follow the French convention of sharp distinction. (That said, I'd feel obliged to explicitly acknowledge the convention, to avoid confusing my audience.)
$\begingroup$Thank you for an informative and helpful answer. The part of the answer that draws on sources, however, doesn't answer the question directly. The question was not about the history of the words per se, but rather the history of how the two concepts I referred to in my question have been denoted, in space. The three citations are silent on this. Perpendicular skew may well be the answer for "orthogonal," but I'd like to see it sourced. And that leaves open whether the other kind should be called perpendicular or perpendicular intersecting, or something else.$\endgroup$
$\begingroup$My wordy anecdote seems to over-power my stated interest in Billingsley's potential role in this. After all, Elements discusses the geometry of three-dimenional figures; I don't know off-hand how or whether (what I've called) "perpendicularly skew" lines are specially named, but they certainly appear, for instance, in the opposite edges of a regular tetrahedron. If Billingsley (or one of this contemporaries) has an opportunity to describe these lines using "his" term "perpendicular", but chooses to reserve the term for coplanar lines, then you have a key data point.$\endgroup$
$\begingroup$[continued] ... But also, given what an astrolabe is for, Chaucer's Treatise likely describes lines in both the plane and space, so his use of "perpendicular" might be particularly insightful. In any case ... My point wasn't to provide mere definitional sources nor to offer a definitive answer, but to provide leads for further investigation, citing specific early places in the mathematical literature in which an author had probably faced the question at hand: What do we call these specially-directed lines in space?$\endgroup$
In geometry books in French, there has traditionally been a very clear distinction between 'orthogonal' lines and 'perpendicular' lines in space.
I suspect that the premise of a traditional distinction between intersecting and nonintersecting orthogonal pairs of lines may be incorrect. The references below have examples from 1900-1921 in textbooks written in English, French and German. Today such a distinction is probably limited to dimension $3$ as presented in some pre-university books, or courses for school teachers.
Problems with the distinction include
it does not work well for parametric families of (pairs of) lines.
in higher dimension there is a clear notion of orthogonality between linear subspaces but it would be complicated to have to judge whether there is an intersection in order to choose the mot juste.
there are too many words like orthocenter, orthologic, orthopole in 2-3 dimensional Euclidean geometry that are incongruous with the idea of orthogonal lines not intersecting. If the orthocenter is the intersection of some orthogonal lines (altitudes) they must be orthogonal to the sides of the triangle. To then alter the language from 2 to 3 dimensions would be strange.
Search results:
1903 UK translation of Franz Hocevar's Solid Geometry book into English has examples of "perpendicular" lines in 3-d being used to include the case of skew lines. Page 10 : "prove that if a straight line be perpendicular to two intersecting lines but does not meet them, it is normal to the plane containing them" and other similar uses on the same page. This was the first old text listed at .
1921 (in USA) Charles Austin Hobbs, Solid Geometry, p271: "in solid geometry, two skew lines are either perpendicular to each other or are oblique to each other".
Rouche's book was in its 7th edition in 1900 and looks like it was a standard text of its time.
The references added to the question support the idea that a distinction between orthogonal and perpendicular is made only in introductory school-books. I guess the rationale is to avoid the possible language confusion for students, between "perpendicular" as a relation between two objects and "the perpendicular" drawn from a point to a line or plane. The perpendicular sounds unique, and is unique. However, if skew lines can be perpendicular, then there are many lines through a point that are perpendicular to a given line, but are not "the" perpendicular to that line.
$\begingroup$I think it is reasonable to assume that until fairly recent times, the basic vocabulary used in solid geometry was unaffected by considerations related to vectors and $n$-dimensional geometry. I would guess until at least 1900 for mathematicians and until 1960 or so in schools. In any event, to the extent that the terminology is used with respect to two lines, the distinction is logical in dimension $n$. What is more likely is that anyone thinking about dimension $n$ is mostly thinking in terms of vectors anyway.$\endgroup$
$\begingroup$Also, if you are uncertain whether there is an intersection, you are free to use "orthogonal," as this includes the "perpendicular" case (much as identical lines are usually called parallel, these days).$\endgroup$
$\begingroup$Updated answer. There is no reason to limit orthogonality to 1-dimensional linear subspaces. All you need is the vector between any pair of points in one subspace to be perpendicular to the vector between any pair in the other subspace. I found some examples in a web search that illustrate why I am suspicious of the premise of the question.$\endgroup$
$\begingroup$Thank you for this answer. I hadn't realized that the French usage I mentioned was actually relatively recent. Your example from Rouché's textbook is what alerted me to this. I've edited my question to include additional details.$\endgroup$
$\begingroup$Perhaps you can determine what the exact terms used in English are when a distinction needs to be made. Do you say "skew perpendicular", "perpendicular skew", "intersecting perpendicular", etc.?$\endgroup$
$\begingroup$Note that the etymology of orthogonal mentions an angle, which assumes a point of intersection; by contrast the etymology of perpendicular only involves direction (any plumb line would be perpendicular to any horizontal line; no point of intersection is required). This is opposite to their supposed distinction in meaning!$\endgroup$
They are tantamount to the same. "Orthogonal" is a term used for more general objects, like planes, whereas "perpendicular" began with, and sticks with lines. As geometry expanded in dimension, so did the definition change. "Orthogonal" would include "Perpendicular" in particular, however, the terms are used synonymously now with no loss of meaning.
$\begingroup$@David I wish I could, but my sources are from what I was taught and not written down (anywhere I can find). Perhaps, it's best to wait for a user to give you a better answer with some evidence, as I can't seem to do that.$\endgroup$
In both cases, vector dot product should vanish. If minimum distance ( along vector cross product) is zero, they are perpendicular (being in the same plane), else skew orthogonal ( there is a minimum skew distance along their common normal).
EDIT1:
We can distinguish between the two or disambiguate between them using a comprehensive 3D picture.
We could bring in 4 points $ P(P_1,P_2), Q(Q_1,Q_2) $ on two lines $P,Q.$
If the volume of tetrahedron spanned by these 4 points ( evaluated by means of well known determinant formula) is zero, then $P$ is perpendicular to $Q$. Else, $P$ is skew orthogonal to $Q.$
$\begingroup$I have no citations per se, but replied more from what I thought would quickly conjure up a mental image of vector contact, based on what I came across commonly. With perpendicularity, the referenced plane is immediately clear; without contact skew situation we are laboring to think about direction of the ( minimum distance) common normal.$\endgroup$ | 677.169 | 1 |
C / C++ / MFC
I got two questions. First I'm making a brick game. I want the ball to bounce off the paddle at different angles depending on where the ball hits the paddle. If the angle variable is specified directly it bounces off in the correct angle but I'm not sure how to calculate the angle depending on where the ball hits the paddle.
I want the ball to bounce off the paddle at different angles depending on where the ball hits the paddle
That is pretty vague, and probably not according to normal physics.
Maybe what you want is: the outgoing angle equals the supplement of the incoming angle plus some delta, which is zero in the center and grows when the hit point is away from the center; so maybe calculate that distance and use it to add to or multiply the outgoing angle.
Cyclone_S wrote:
going in a circle
the equations for a circle in two dimensional space are:
(x - xc)^2 + (y - yc)^2 = r^2
or
x = xc + r * cos(a)
y = yc + r * sin(a)
where (x,y) is a point on the circle, (xc,yc) is the center, r the radius, a an angle in radians.
I cannot believe you would not know that. Look at the equations, they say the point(x,y) is at a fixed distance r from a fixed point (xc,yc).
The quality and detail of your question reflects on the effectiveness of the help you are likely to get. Please use <PRE> tags for code snippets, they improve readability. CP Vanity has been updated to V2.3
Assuming he is adolescent or older, he would have learned this and much more at school, both as formula's for describing a circle, and as a geometric illustration for explaining what a sine and cosine actually are. He already was using angles, sine and cosine, in his post, so it puzzles me how he would not come up with the equations if he had ever seen and understood themThanks for the replies. Both problems are mostly solved. I have the paddle/ball equation figured out but I'm having a problem where any value less then 1 is ignored... any ideas? I need finer precision. ThanksThe quality and detail of your question reflects on the effectiveness of the help you are likely to get. Please use <PRE> tags for code snippets, they improve readability. CP Vanity has been updated to V2.4
You might want to check out this site[Physics engines for dummies] for an easy introduction into the basics of vector algebra used to emulate quasi-physical simulations. You will find that in stead of angles and sinus/cosinus it explains the calculations needed for reflections with the help of vector algebra. Although the code is not C/C++ it should be easy enough to translate the relevant code for your problems.
In practice, term "Garbage Collector" usually implies non-deterministic cleanup and that rules out reference counting. In other words, with (automatic) reference counting there is no garbage to be collected: an object gets released at the moment the last reference to it goes out of scope.
GC is a form of global memory management that keeps track of memory resources and returns freed memory at undefined intervals. For performance reasons, GC often does not immediately return freed memory, but instead just marks it as freed later.
RC is a method to locally ensure that an object is freed as soon as it isn't needed anymore. It determines this by keeping count of the references to the object that have been handed out.
In a way you coud consider RC as a very localized GC. Also, as has been pointed out, RC is often used to implement a GC and find out which memory blocks may be released.
GC has better overall performance, but may lead to heavier use of memory. Using only RC keeps the memory footprint at a minimum, but is somewhat slowerUnicode is a vast specification covering many things. Encluding serveral representations of some very broad (many natural languages) character sets.
There are also associated encodings for those character sets.
Some encodings have a fixed size and some are variable.
A variable size encoding has some byte sequences which represent specific characters and other byte sequences which are used as flags to indicate that additional bytes are needed to determine the actual character.
A multi-byte character set might either mean a fixed size representation of a character set but normally means a variable sized encoding which started with a single byte for the initial encoding. UTF8 is a variable sized encoding and can thus also be considered a multibyte character set.
I have to develop an application that captures a portion of the desktop every 30 seconds, after viewing the captured image in the window of application, should find (and then highlight) which regions of the new image have changed compared to the previous image. | 677.169 | 1 |
A locus (in geometric terms) is a series of points that is determined by specific conditions. When we work with loci in math, it makes me feel like I am placing a pin in corkboard. We place that pin based on the instructions that are provided. In most cases, we will just moving up and down the y-axis or left and right across the x-axis. In these worksheets you will be the one placing the pin in the corkboard.
What is a locus at a fixed distance? A locus is the arrangement of all focuses which fulfill a specific condition. The locus at a fixed separation, d, from point P is a hover with the given point P as its inside and d as its span. The locus at a fixed separation, d, from a line m, is a couple of equal lines a good way off of d from line m and situated on either side of m.
The locus is equidistant from two focuses. An and B is the opposite bisector of the line fragment joining the two focuses. The locus equidistant from two parallel lines, m1 and m2, is a line corresponding to both m1 and m2 and somewhere between them.
These worksheets explain how to find the locus of two points at a fixed distance and write its equation.
Students will describe the locus indicated. Example problem: A wooden block is 200 feet long and 40 feet wide. It is planned to cut it 28 feet
from the center of the block. Describe where it will be cut. Ten problems are provided.
Students review how to find these measures. Here is a sample problem: A book-cover is 12 feet long and 6 feet wide. It is planned to color it 5 foot
from the center of the book-cover. Describe where it will be painted. Six practice problems are provided.
Students will demonstrate their proficiency this skill. Example: A railway track is 125 feet long and 50 feet wide. It is planned to cover it with
mud 30 feet from the center of the track. Describe where it will be covered
with mud. Ten problems are provided.
Students will find the locus described and write its equation. Example: Describe the locus of points 6 units from
the line y = -14. Three problems are provided, and space is included for students to copy the correct answer when given.
Example: A tree A is 50 feet from another tree B. Shadowing range of A is 30 feet and that
of B is 25 feet. Draw a diagram showing the areas where each tree shadows. Is
any area shadowed by both the trees?
A workshop is located at the coordinates (5, 12) on a coordinate grid. The
delivery service extends for 5 km. Write the equation of the locus which
represents the outer edge of the delivery service area. | 677.169 | 1 |
Radian/Nanosecond Converter
What Unit of Measure is Radian/Nanosecond?
Radian per nanosecond is a unit of measurement for angular velocity. By definition, one radian per nanosecond represents change in the orientation of an object by one radian every nanosecond.
What is the Symbol of Radian/Nanosecond?
The symbol of Radian/Nanosecond is rad/ns. This means you can also write one Radian/Nanosecond as 1 rad/ns.
Manually converting RadianNanosecond converter tool to get the job done as soon as possible.
We have so many online tools available to convert Radian/Nanosecond to other Angular Velocity units, but not every online tool gives an accurate result and that is why we have created this online Radian/Nanosecond converter tool. It is a very simple and easy-to-use tool. Most important thing is that it is beginner-friendly.
How to Use Radian/Nanosecond Converter Tool
As you can see, we have 2 input fields and 2 dropdowns. For instance, you want to convert Radian/Nanosecond to Radian/Minute.
From the first dropdown, select Radian/Nanosecond and in the first input field, enter a value.
From the second dropdown, select Radian/Minute.
Instantly, the tool will convert the value from Radian/Nanosecond to Radian/Minute and display the result in the second input field.
Example of Radian/Nanosecond Converter Tool
Radian/Nanosecond
1
Radian/Minute
60000000000
Radian/Nanosecond to Other Units Conversion Table
Conversion
Description
1 Radian/Nanosecond = 57295779513.08 Degree/Second
1 Radian/Nanosecond in Degree/Second is equal to 57295779513.08
1 Radian/Nanosecond = 57295779.51 Degree/Millisecond
1 Radian/Nanosecond in Degree/Millisecond is equal to 57295779.51
1 Radian/Nanosecond = 57295.78 Degree/Microsecond
1 Radian/Nanosecond in Degree/Microsecond is equal to 57295.78
1 Radian/Nanosecond = 57.3 Degree/Nanosecond
1 Radian/Nanosecond in Degree/Nanosecond is equal to 57.3
1 Radian/Nanosecond = 3437746770784.9 Degree/Minute
1 Radian/Nanosecond in Degree/Minute is equal to 3437746770784.9
1 Radian/Nanosecond = 206264806247100 Degree/Hour
1 Radian/Nanosecond in Degree/Hour is equal to 206264806247100
1 Radian/Nanosecond = 4950355349930300 Degree/Day
1 Radian/Nanosecond in Degree/Day is equal to 4950355349930300
1 Radian/Nanosecond = 34652487449512000 Degree/Week
1 Radian/Nanosecond in Degree/Week is equal to 34652487449512000
1 Radian/Nanosecond = 150676440963500000 Degree/Month
1 Radian/Nanosecond in Degree/Month is equal to 150676440963500000
1 Radian/Nanosecond = 1808117291562000000 Degree/Year
1 Radian/Nanosecond in Degree/Year is equal to 1808117291562000000
1 Radian/Nanosecond = 1000000000 Radian/Second
1 Radian/Nanosecond in Radian/Second is equal to 1000000000
1 Radian/Nanosecond = 1000000 Radian/Millisecond
1 Radian/Nanosecond in Radian/Millisecond is equal to 1000000
1 Radian/Nanosecond = 1000 Radian/Microsecond
1 Radian/Nanosecond in Radian/Microsecond is equal to 1000
1 Radian/Nanosecond = 60000000000 Radian/Minute
1 Radian/Nanosecond in Radian/Minute is equal to 60000000000
1 Radian/Nanosecond = 3600000000000 Radian/Hour
1 Radian/Nanosecond in Radian/Hour is equal to 3600000000000
1 Radian/Nanosecond = 86400000000000 Radian/Day
1 Radian/Nanosecond in Radian/Day is equal to 86400000000000
1 Radian/Nanosecond = 604800000000000 Radian/Week
1 Radian/Nanosecond in Radian/Week is equal to 604800000000000
1 Radian/Nanosecond = 2629800000000000 Radian/Month
1 Radian/Nanosecond in Radian/Month is equal to 2629800000000000
1 Radian/Nanosecond = 31557600000000000 Radian/Year
1 Radian/Nanosecond in Radian/Year is equal to 31557600000000000
1 Radian/Nanosecond = 63661977236.76 Gradian/Second
1 Radian/Nanosecond in Gradian/Second is equal to 63661977236.76
1 Radian/Nanosecond = 63661977.24 Gradian/Millisecond
1 Radian/Nanosecond in Gradian/Millisecond is equal to 63661977.24
1 Radian/Nanosecond = 63661.98 Gradian/Microsecond
1 Radian/Nanosecond in Gradian/Microsecond is equal to 63661.98
1 Radian/Nanosecond = 63.66 Gradian/Nanosecond
1 Radian/Nanosecond in Gradian/Nanosecond is equal to 63.66
1 Radian/Nanosecond = 3819718634205.5 Gradian/Minute
1 Radian/Nanosecond in Gradian/Minute is equal to 3819718634205.5
1 Radian/Nanosecond = 229183118052330 Gradian/Hour
1 Radian/Nanosecond in Gradian/Hour is equal to 229183118052330
1 Radian/Nanosecond = 5500394833255900 Gradian/Day
1 Radian/Nanosecond in Gradian/Day is equal to 5500394833255900
1 Radian/Nanosecond = 38502763832791000 Gradian/Week
1 Radian/Nanosecond in Gradian/Week is equal to 38502763832791000
1 Radian/Nanosecond = 167418267737230000 Gradian/Month
1 Radian/Nanosecond in Gradian/Month is equal to 167418267737230000
1 Radian/Nanosecond = 2009019212846700000 Gradian/Year
1 Radian/Nanosecond in Gradian/Year is equal to 2009019212846700000
1 Radian/Nanosecond = 63661977236.76 Gon/Second
1 Radian/Nanosecond in Gon/Second is equal to 63661977236.76
1 Radian/Nanosecond = 63661977.24 Gon/Millisecond
1 Radian/Nanosecond in Gon/Millisecond is equal to 63661977.24
1 Radian/Nanosecond = 63661.98 Gon/Microsecond
1 Radian/Nanosecond in Gon/Microsecond is equal to 63661.98
1 Radian/Nanosecond = 63.66 Gon/Nanosecond
1 Radian/Nanosecond in Gon/Nanosecond is equal to 63.66
1 Radian/Nanosecond = 3819718634205.5 Gon/Minute
1 Radian/Nanosecond in Gon/Minute is equal to 3819718634205.5
1 Radian/Nanosecond = 229183118052330 Gon/Hour
1 Radian/Nanosecond in Gon/Hour is equal to 229183118052330
1 Radian/Nanosecond = 5500394833255900 Gon/Day
1 Radian/Nanosecond in Gon/Day is equal to 5500394833255900
1 Radian/Nanosecond = 38502763832791000 Gon/Week
1 Radian/Nanosecond in Gon/Week is equal to 38502763832791000
1 Radian/Nanosecond = 167418267737230000 Gon/Month
1 Radian/Nanosecond in Gon/Month is equal to 167418267737230000
1 Radian/Nanosecond = 2009019212846700000 Gon/Year
1 Radian/Nanosecond in Gon/Year is equal to 2009019212846700000
1 Radian/Nanosecond = 159154943.09 Revolution/Second
1 Radian/Nanosecond in Revolution/Second is equal to 159154943.09
1 Radian/Nanosecond = 159154.94 Revolution/Millisecond
1 Radian/Nanosecond in Revolution/Millisecond is equal to 159154.94
1 Radian/Nanosecond = 159.15 Revolution/Microsecond
1 Radian/Nanosecond in Revolution/Microsecond is equal to 159.15
1 Radian/Nanosecond = 0.1591549430919 Revolution/Nanosecond
1 Radian/Nanosecond in Revolution/Nanosecond is equal to 0.1591549430919 | 677.169 | 1 |
Power of a point — Figure 1. Illustration of the power of point P in the circle centered on the point O. The distance s is shown in orange, the radius r is shown in blue, and the tangent line segment PT is shown in red. In elementary plane geometry, the power of a… … Wikipedia
Power center (geometry) — The radical center (orange point) is the center of the unique circle (also orange) that intersects three given circles at right angles. In geometry, the power center of three circles, also called the radical center, is the intersection point of… … Wikipedia
Point process — In statistics and probability theory, a point process is a type of random process for which any one realisation consists of a set of isolated points either in time or geographical space, or in even more general spaces. For example, the occurrence | 677.169 | 1 |
How many Parallel Sides can a Triangle Have
It is fascinating to learn How many Parallel Sides can a Triangle Have. Triangle is an important civil engineering geometry structure. And is a geometric entity from the family of polygons that line in a closed shape. To understand the possibilities of parallel sides in a triangle, we need to understand a variety of situations.
It is a fact that no side of a triangle is parallel to any other side of the same triangle. However, there are other cases or situations that we will take a look at in this article. The first situation is of a single triangle which is the case in every mathematics example. A triangle is a closed body with three sides. The lines of a triangle are meant to connect, end-to-end. And whenever these lines connect, that conclusively means that they are crossing one another. You can learn more about the geometry of triangle in this article.
How many Parallel Sides can a Triangle Have
So, there couldn't be any parallel sides in a triangle. A pair of lines are Parallel if only they are at the same angle and never collide. Additionally, some theorems introduce a fourth line parallel to any side of a triangle, but that doesn't concern us. It interests us to talk about triangles alone. Take a look at image below, they consist of two similar triangles of different sizes. However, these triangles have sides parallel to one another.
triangle with parallel sides
The above triangle shows a pattern where infinite independent triangles with identical angles can have parallel sides. All these sides are parallel in one way only, that is if we do not produce any side like an array. However, if we produce or elongate any side considering it an array, that will introduce collision of all sides.
There is another situation with identical results and you can have a look at the image below. These triangles a and b are identical and have all sides parallel to their corresponding sides.
How many Parallel Sides can a Triangle Have
There is another situation, but that doesn't come under the laws of mathematics. It is based on a fictitious design of triangles. Have a look at the image below. These stairs form an infinite loop of triangles that don't have common end-points where they form a closed shape. And in these open end illusion-triangles there are sides parallel to their corresponding sides under or above.
triangular stairs
There were various examples under discussion and it helped conclude the possibilities of parallel sides of triangles.
There is one case that confuses people about triangles having parallel sides. And that is a polygon, it is due to a theorem that discusses an imaginary condition where a parallel line cuts two sides of a triangle. Take a look at the image below.
triangle with a parallel side
The image above has three sides a, b, and c. All these lines cross one another at three corners and are not parallel in any case. However, line A that starts from side 1 and ends at side 2 is parallel to side 3. This bisecting line is a case where with the addition of an external line in a triangle, one can have a side of a triangle parallel to a 4th side that changes its geometry | 677.169 | 1 |
Draw a ΔABC in which BC = 6 cm, AB = 4 cm and AC = 5 cm.
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Draw a ΔABC in which BC = 6 cm, AB = 4 cm and AC = 5 cm. Draw a triangle similar to ΔABC with its sides equal to (3/4)th of the corresponding sides of ΔABC.
Solution:
Given that
Construct a triangle of sides $A B=4 \mathrm{~cm}, B C=6 \mathrm{~cm}$ and $A C=5 \mathrm{~cm}$ and then a triangle similar to it whose sides are $(3 / 4)^{\text {th }}$ of the corresponding sides of $\triangle A B C$.
We follow the following steps to construct the given
Step of construction
Step: I- First of all we draw a line segment.
Step: II- With A as centre and radius, draw an arc.
Step: III -With B as centre and radius, draw an arc, intersecting the arc drawn in step II at C. | 677.169 | 1 |
How do you find RAA?
Category: sciencespace and astronomy
4.7/5(2,638 Views . 32 Votes)
If the terminal arm is in quadrant 2, do 180∘ minus the principle angle to find the related acute angle. If the terminal arm is in quadrant 3, do the principle angle minus 180∘ to find the related acute angle. If the terminal arm is in quadrant 4, do 360∘ minus the principle angle to find the related acute angle.
Thereof, how do you find the reference angle on a calculator?
Choose a proper formula for calculating the reference angle:
0° to 90°: reference angle = angle ,
90° to 180°: reference angle = 180° - angle ,
180° to 270°: reference angle = angle - 180° ,
270° to 360°: reference angle = 360° - angle .
Secondly, what is an example of an acute angle? A right angle is an angle that measures 90 degrees. An acute angle is any angle smaller than a right angle. An example of an acute angle is a pizza slice.
Thereof, what is a real life example of an acute angle?
Some real-life examples of acute angles are:The arms of a wall clock make acute angles at several hours of a day. For example, 2 o' Clock. The hour hand and the minute hand forming an acute angle at 2 o' Clock.
What is the reference angle of?
Reference Angle. When an angle is drawn on the coordinate plane with a vertex at the origin, the reference angle is the angle between the terminal side of the angle and the x-axis. The reference angle is always between 0 and 2π radians (or between 0 and 90 degrees).
Coterminal Angles are angles who share the same initial side and terminal sides. Finding coterminal angles is as simple as adding or subtracting 360° or 2π to each angle, depending on whether the given angle is in degrees or radians. There are an infinite number of coterminal angles that can be found.
Reference Angles. A reference angle for a given angle in standard position is the positive acute angle formed by the $x$-axis and the terminal side of the given angle. Reference angles, by definition, always have a measure between 0 and .
Standard Position of an Angle - Initial Side - Terminal Side. An angle is in standard position in the coordinate plane if its vertex is located at the origin and one ray is on the positive x-axis. The ray on the x-axis is called the initial side and the other ray is called the terminal side.
An angle in standard position is determined by a counter- clockwise rotation and is always positive. An angle determined by a clockwise rotation is always negative. If the terminal arm of the principal angle lies in quadrant 2 then the related acute angle is calculated as β = 180° - θ.
A Quadrantal Angle is an angle in standard position with terminal side on the x-axis or y-axis. Some examples are the angles located at 0°, 90°, 180°, 270°, 360°, 450°, as well as -90°, -180°, -270°, -360°
Cotangent, Secant and Cosecant. Cosecant is the reciprocal of sine. Secant is the reciprocal of cosine. Cotangent is the reciprocal of tangent. When solving right triangles the three main identities are traditionally used. | 677.169 | 1 |
Python Program to Calculate Area of any Triangle using its Coordinates
Enhancing programming skills is very important no matter what language you have chosen. So, practice frequently with these simple java programs examples and excel in coding the complex logic.
GIven three coordinates of a triangle the task is to find the area of the given Triangle in Python.
Examples:
Example1:
Input:
x coordinate of the first point = 6
y coordinate of the first point = 2
x coordinate of the second point = 9
y coordinate of the first point = 11
x coordinate of the third point = 5
y coordinate of the first point = 17
Output:
The Area of the triangle with the given coordinates (6,2) (9,11) (5,17) = 27.0
Example2:
Input:
x coordinate of the first point = 3
y coordinate of the first point = 7
x coordinate of the second point = 2
y coordinate of the first point = 11
x coordinate of the third point = 9
y coordinate of the first point = 7
Output:
The Area of the triangle with the given cordinates (3,7) (2,11) (9,7) = 12.0
Program to Calculate Area of any Triangle using its Coordinates in
Python
Below are the ways to calculate to find the area of the given Triangle in Python.
Let the coordinates of a triangle be A(x1, y1), B(x2, y2), and C(x3, y3). Using the Mathematical formula, we can compute the area of triangle ABC.
Area of Triangle = |(1/2)*(x1*(y2-y3)+x2*(y3-y1)+x3*(y1-y2))|
Calculating the Area without considering the modulus may result in a negative value. We only take the magnitude by applying modulus to the expression because Area cannot be negative.
In the program, we utilize the abs() method to get the absolute value or magnitude.
Method #1: Using Mathematical Formula (Static Input)
Approach:
Give the 3 coordinates as static input and store static input and store them in 6 separate variables.
xcor1 = 3
ycor1 = 7
xcor2 = 2
ycor2 = 11
xcor3 = 9
ycor3 = 7 ")The Area of the triangle with the given cordinates (3,7) (2,11) (9,7) = 12.0
Method #2: Using Mathematical Formula (User Input)
Approach:
Give the 3 coordinates as user input using map(),int(),split() functions.
Store user input using map(),int(),split() functions.
# Store them in 6 separate variables.
xcor1 = int(input('Enter some random x coordinate of the first point = '))
ycor1 = int(input('Enter some random y coordinate of the first point = '))
xcor2 = int(input('Enter some random x coordinate of the second point = '))
ycor2 = int(input('Enter some random y coordinate of the first point = '))
xcor3 = int(input('Enter some random x coordinate of the third point = '))
ycor3 = int(input('Enter some random y coordinate of the first point = '))")Enter some random x coordinate of the first point = 6
Enter some random y coordinate of the first point = 2
Enter some random x coordinate of the second point = 9
Enter some random y coordinate of the first point = 11
Enter some random x coordinate of the third point = 5
Enter some random y coordinate of the first point = 17
The Area of the triangle with the given coordinates (6,2) (9,11) (5,17) = 27.0 | 677.169 | 1 |
do all triangles equal 180 degrees
Why do Angles in a Triangle Add to 180 Degrees? - Answered by a verified Tutor. We're going to do exactly the same in proving that the sum of the angles in a triangle is 180 degrees. Before we get too far into our story about triangles and the total number of degrees in their three angles, there's one little bit of geometric vocabulary that we should talk about. Given that three angles of a triangle are represented by 2x, x+6, and 3x, what are the measures of those angles if the sum of three angles of a triangle are always equal to 180 degrees? Each corner that you cut off contains an angle from the triangle. If so, your picture should look like this: What's the point of this picture? In short, the interior angles are all the angles within the bounds of the triangle. Equilateral: "equal"-lateral (lateral means side) so they have all equal sides; Isosceles: means "equal legs", and we have two legs, right? 1 decade ago. 2x + 45 = 180 . As it turns out, you can figure this out by thinking about the interior and exterior angles of a triangle. Isosceles: means \"equal legs\", and we have two legs, right? Just remember that the interior angles of a triangle ALWAYS add up to 180 degrees. spherical geometry. Types of Triangles - Cool Math has free online cool math lessons, cool math games and fun math activities. Why 180 and not some other number? Therefore, straight angle ABD measures 180 degrees. And our little drawing shows that the exterior angle in question is equal to the sum of the other two angles in the triangle. And do all triangles really contain 180 degrees? This one's y. In Reimannian Geometry: No, they actually add up to more than 180 degrees. The lesson presents them with a theory that all angles of ANY triangle when added will equal 180 degrees. Anna, a 5th grader: "The sum of the angles of a triangle is not always 180 o !" The other one is an isosceles triangle that has 2 angles measuring 45 degrees (45–45–90 triangle). Draw a perpendicular from the vertex opposite the longest side, to the longest side. What Type of Angle? Please explain why all triangles equal up to 180 degrees. 2x = 135. x = 67.5. This rectangle has four 90 degree angles adding up to 360 degrees. x + x + 45 = 180. And triangles also have a lot to do with rectangles, pentagons, hexagons, and the whole family of multi-sided shapes known as polygons. Lv 7. 0 0. Scalene: means \"uneven\" or \"odd\", so no equal sides. Quick & Dirty Tips™ and related trademarks appearing on this website are the property of Mignon Fogarty, Inc. and Macmillan Publishing Group, LLC. :D Must be credible. this has point symmetry. Jason Marshall is the author of The Math Dude's Quick and Dirty Guide to Algebra. How about that! In a small triangle on the face of the earth, the sum of the angles is only slightly more than 180 degrees. Pick any two longitudinal lines, extend them from the North Pole to the Equator, and presto! However, three successive rotations around the vertices of a triangle do not necessarily cause a line to rotate four right angles! The measure of the interior angles of the triangle, x plus z plus y. As an example, here's another one that I've made: The inevitable conclusion of this game is that the interior angles of a triangle must always add up to 1800. In Euclidean Geometry: If it is a triangle with 3 sides yes it does. This tutorial shows you how to put this knowledge into an equation and solve to find that missing measurement! We gave two proofs of the Pythagorean theorem. 2x = 180 - 45. The measure of this angle is x. Suppose you triangle is not a right triangle. To explore the truth of this rule, try Math Warehouse's interactive triangle , which allows you to drag around the different sides of a triangle and … Just like regular numbers, angles can be added to obtain a sum, perhaps for the purpose of determining the measure of an unknown angle. We're going to do exactly the same in proving that the sum of the angles in a triangle is 180 degrees. This is why we coloured the edges so we can easily see the angle contained by the edges. Check this link for reference: "In Depth Analysis of Triangles on Sphere" and "Friendly intro to Triangles on Sphere." The unknown third angle is easy to find since we know that all of them added up will equal 180. This problem is a triangle. But what? Might there be some limitation to our drawing that is blinding us to some other more exotic possibility? In Euclidean geometry, the sum of the interior angles of a triangle is always exactly 180° as proved by Euclid in his Elements Book I, proposition 32. A triangle with all interior angles measuring less than 90° is an acute triangle or acute-angled triangle. Which brings us to the main question for today: Why is it that the interior angles of a triangle always add up to 1800? Since the triangles are congruent each triangle has half as many degrees, namely 180. In other words, the other two angles in the triangle (the ones that add up to form the exterior angle) must combine with the angle in the bottom right corner to make a 1800 angle. Proof that a Triangle is 180 Degrees One of the first things we all learned about triangles is that the sum of the interior angles is 180 degrees. But for today, we're going to start by figuring out exactly why it is that the angles of a triangle always add up to 1800. Triangles can also have names that tell you what type of angle is inside: All triangles have internal angles that add up to 180°, no matter the type of triangle. The second followed Euclid and was more technical. The answer is 'sometimes yes, sometimes no'. So you have 90 minus theta plus 90 degrees plus 32 degrees-- so I'm going to do that in a different color-- is going to be equal to 180 degrees. Triangles that do not have an angle measuring 90° are called oblique triangles. Mr. Cohen told me that the number of degrees in a full turn or circle, is 360'. Angle DBA is equal to CAB because they are a pair of alternate interior angle. Try making a few drawings starting with different triangles of your choosing to see this for yourself. That's all we're doing over here. Or so you thought … because we're also going to see that sometimes they don't. All isosceles triangles: - Have angles that add up to 180 degrees - Have two equal sides. Is it a meaningful question? To see what I mean, either grab your imagination or a sheet of paper because it's time for a little mathematical arts-and-crafts drawing project. Already know the other two interior angle measurements? Répondre Enregistrer. Non-Euclidean geometry is concerned with geometry that isn't on a flat plane such as the globe and is used mostly in advanced physics and mathematics such as in the general theory of relativity. We've learned that triangles are unique shapes with the interesting fact that all of the angles added up will always equal 180 degrees. If you think about it, you'll see that when you add any of the interior angles of a triangle to its neighboring exterior angle, you always get 1800—a straight line. Solution : Let us add all the three given angles and check whether the sum is equal to 180 °. 1st place: youl'll get 2 servings of spanish food 2nd place: you'll get 2 cookies 3rd place: you'll get a bottle of water and a The unequal side is called the base. Triangle ABC is congruent to triangle A'B'C' so the bow-tie shaped shaded area, marked Area 2, which is the sum of the areas of the triangles ABC and A'BC', is equal to the area of the lune with angle B, that is equal to 2B.. When we assemble the angles (by aligning the coloured edges), we see that all the angles add up to a straight line (or 180°). If you have a protractor handy, it'd be great to measure and add up the triangle's interior angles and check that they're pretty close to 1800. We use cookies to give you the best possible experience on our website. Once an angle is larger than 180 degrees, it is categorized as a reflex angle. In other words, they're the kind of angles we've been talking about all along. As we know, if we add up the interior and exterior angles of one corner of a triangle, we always get 1800. If we move these angles so that they fit in next to each other, we can see that they rest on a straight line. Write the equation angle PAB + angle BAC + angle CAQ = 180 degrees. But we've just completed our proof. The sum of the angles of a spherical triangle is not equal to 180°. Why is that? Since today's theme is the triangle, let's talk about the interior and exterior angles of a triangle. Suppose you triangle is not a right triangle. Procure an uninflated balloon, lay it on a flat surface, and draw as close to as perfect of a triangle on it as you can. Therefore, a complete rotation is 360 degrees. You have already applied the triangle sum theorem which states that all 3 angles in a triangle add up to #180# degrees it seems, so now all you have to do is create an algebraic expression that reflects that. Interior and exterior angles of all triangles is x and take a look at the bottom of..., 2, none: 1 3 = 180° why is every triangle 180 degrees protractor... That it is a visual proof that the sum of the angles in any triangle equals degrees... To next week 's article where we started exploring the strange and wonderful world known as non-Euclidean geometry,. Know how the angles of a square/rectangle, the two angles are 67.5 degrees equals 180° but. To either prove this is correct or incorrect to some other more exotic possibility you can add four!, is 360 ' contain 360 degrees once again to sum up its interior angles a. A plane! how a triangle is that it is common knowledge that the exterior angle in a so. 67.5 degrees angle measurement in a right triangle: 1 whether the sum of angle... A reflex angle are three different types of triangles - cool math lessons, cool math games and fun activities! A spherical triangle is exactly 180° ( = π ) three different types of basic triangles three given angles check! Those pedantic folks, I mean flat triangles on a straight line and 180... Triangle ) sides to equal 360 to do exactly the same in that... The three angles in a triangle do not necessarily cause a line to rotate right. The expanding nature of mathematics sides to equal 360 are good approximations triangles together they form a square or.! Is blinding us to some other more exotic possibility Google Chrome or Firefox and 45 degrees ( 45–45–90 ). + angle 3 = 180° why is every triangle 180 degrees a perpendicular from North! Also add up to 180 degrees we always get 1800 Equator at 90 degree angles,... Exactly the same in proving that the sum of the three angles in a rectangle CAB! 60° and 90° be the angles of one corner of a triangle is half of spherical! Can use this to find out what the angles of the interior angles and! About all along of one corner of a triangle that the sum of the earth, the degrees 180. Can be 3, 2, none: 1 to listen to the sum of the three angles. Flat surface ) which is what most people will always equal 180 degrees add angles true: https //shortly.im/0V2Py... In proving that the number of degrees in total see exactly what mean! '' joined by an \ '' Sides\ '' joined by an `` Odd '' side so you thought because... Expanding nature of mathematics work in small groups to either prove this correct. '' Sides\ '' joined by an `` Odd '' side out of cards thus the... 90° are called acute to equal 360 ( 90 degree angles ), can... Marshall is the triangle the result is a half-circle a half-circle triangle ( the one ". So you thought … because we know that line and measures 180 degrees '! Yes it does in small groups to either prove this is why we the! You: Parent or acute-angled triangle geometries exist, for which we know that all of angles. Is common knowledge that the sum of the three angles in a full turn or,. That you cut off contains an angle measuring 90° are called acute ( )... Wonderful world known as non-Euclidean geometry the same reasoning goes with the interesting fact that all them... Equal are congruent each triangle has a half of a rectangle that add up the and! The 3 interior angles of a rectangle so those two triangles have internal that... Coloured the edges and 180 degrees, it is no longer true the! We hope, convinced you that the Pythagorean Theorem is true all similar triangles congruent. As it turns out, you can figure this out by thinking about the interior of! One corner of a triangle experience on our website `` the sum is 180 degrees ( 90 degree angles up... Rotate four right angles Bookshop.org Affiliate, QDT earns from qualifying purchases to do with a single straight line up... Acute triangle or acute-angled triangle quick refresher: there are three different types triangles. To investigate the sum of the three angles of a square/rectangle, the sum of the original triangle ( one... Their angles must sum to 180° the original triangle ( the one labeled " a " ) have... Add all four sides to equal 360 the interior do all triangles equal 180 degrees of one corner of a for! Triangle or acute-angled triangle they do n't ) so they have all equal sides the unknown do all triangles equal 180 degrees angle is to! Which, by the above contain 360 degrees three different types of basic triangles of angles the. ) which is what most people will always equal 180 degrees but cant find valid. I use the pump to inflate the globe and show how a.. Congruent triangles together they form a square or retangle triangle 180 degrees to very much more than 180.! Four sides to equal 360 forms a straight line must be so has two equal \ '' Odd\ ''....: no, so it can also be called an equiangular triangle why do angles in rectangle... Alphabetical order then he asked me if I knew about the sum of the angles a... Acute-Angled triangle in Depth Analysis of triangles - cool math has free online cool games. And 95° be the angles added up will equal 180 in proving that the sum of the Dude! Surface, but how do we know that the sum of the three angles of a triangle add! Ll see exactly what I mean by this over the next few weeks are different than... On our website is no longer true that the sum must be equal to the longest side a! I mean flat triangles on sphere '' and `` Friendly intro to triangles on a sphere have! As an Amazon Associate and a Bookshop.org Affiliate, QDT earns from qualifying purchases to 1800 goes the! Https: //shortly.im/0V2Py to find a missing angle in a triangle than 180° if all sides are equal, no... Pab, angle BAC + angle 3 = 180° why is every triangle 180 degrees are congruent each has! \ '' Odd\ '', so no equal sides the unknown third angle is to... Surface, but locally the laws of the angles of a triangle always add to! A few drawings starting with different triangles of your choosing to see this for.! Acute triangle or acute-angled triangle to sum up its interior angles of a triangle 's side directly its. D forms a straight line with vertices a, B, and originate from vertex. Plane triangle is not equal to 180° and any rectangle has the sum of its angles sake of,! Yes, sometimes no ' for reference: `` the sum is 180 ) acute triangle or acute-angled triangle measuring... Equals 180°, but locally the laws of the angles of a triangle to! A long time whether other geometries exist, for which we know, if we can determine a angle! The expanding nature of mathematics here ' s the point of this picture the opposite. Or curiosity may have crept in ) which is what most people will always deal with are exactly degrees. The equation angle PAB, angle BAC, and we have two equal sides result!: //shortly.im/0V2Py to understanding why ( and when ) the angles of triangle. Thibaut 's argument 360 degrees is the author of the triangles are unique shapes with the interesting that. Let us add all the angles of one corner of a triangle do necessarily... Between these two things is with an example same in proving that the sum of the added! We always get 1800 these two things is with an example see that sometimes they do.... Basic triangles then he asked me if I knew about the sum of the given! And 180 degrees all angles will be equal we could just reorder this if add... Locally the laws of the three angles right geometry are good approximations think about try. Pump to inflate the globe and show how a triangle is ∘ which we know if. To participate in the world does a triangle is that it is no longer true that number. To 1800 look like this ending to the Equator at 90 degree angles ), can... They do n't is that it is a visual proof that the sum of the interior angles EBC and.. ) so they have all equal sides and we have two legs, right is 180. Why all triangles have internal angles that add up the interior and exterior angles a! One was short and, we hope, convinced you that the sum of the angle inside of a equal! Up the interior and exterior angles of a triangle is ∘, and we have two angles be. It turns out, you can create a triangle a and C respectively: have... The best possible experience on our website sum up its interior angles EBC and ACB this little! How to put this knowledge into an equation and solve to find since we know that all angles be! Yes it does the laws of the angles in a small triangle on the face of the angles are! An isosceles triangle will have a triangle equal 180 degrees lunes with angles and. Length of a triangle add up to more than 180 degrees but cant find a angle. Would like to listen to the Equator at 90 degree angles ), you add! Person Who wants to participate in the triangle, we ' ll see exactly what I by. | 677.169 | 1 |
2 Shapes and symmetry
2.1 Geometric shapes – triangles
This section deals with the simplest geometric shapes and their symmetries. All of the shapes are two-dimensional – hence they can be drawn accurately on paper.
Simple geometric shapes are studied in mathematics partly because they are used in thousands of practical applications. For instance, triangles occur in bridges, pylons and, more mundanely, in folding chairs; rectangles occur in windows, cinema screens and sheets of paper; while circles are an essential part of wheels, gears and plates.
By definition, triangles are shapes with three straight sides. However, there are various types of triangle:
An equilateral triangle is a triangle with all three sides of equal length. The three angles are also all equal.
An isosceles triangle is a triangle with two sides of equal length. The two angles opposite the equal sides are also equal to one another.
A right-angled triangle is a triangle with one angle that is a right angle.
A scalene triangle is a triangle with all the sides of different lengths. The angles are also all different.
It is a general convention that equal sides are marked by drawing a short line, /, through them, and a right angle is marked by a square between the arms of the angle. If sides and angles are not marked, do not assume that they are equal, just because they look equal | 677.169 | 1 |
Page 56 ... parallelogram is a quadrilateral which has site sides parallel . Trapezium . Trapezoid . Parallelogra 169. A rectangle is a parallelogram which has right angles . 170. A rhomboid is a parallelogram which has oblique angles . 171. A ...
Page 57 ... parallelogram or trapezoid is the perpendicular distance between its bases . 177. The diagonal of a quadrilateral is a straight line joining two opposite vertices . PROPOSITION XXXVII . THEOREM . 178. The diagonal of a parallelogram | 677.169 | 1 |
Chapter 11 of RD Sharma's Class 9 Maths Solutions
Explanation of each exercise in detail, including a list of important topics, will be there in this solution.
While studying for the final Class 9 Maths exam, a student must remember to verify all of the important aspects of Class 9 Chapter 11 RD Sharma Solutions. The students will learn how to solve the crucial mathematical questions in Chapter 11 with the help of RD Sharma solutions.
An in-depth understanding of the parts that come under Coordinate Geometry can help you secure good marks. All of the important matter is covered in Chapter 11 of the RD Sharma Class 9 Solutions. The key is to practise regularly, and we've attempted to clarify the exercise-by-exercise explanations here. In this exercise, students will learn -
Chapter 11 Exercise 11.2 RD Sharma Solutions for Class 9 Maths
Construction of varied shapes is important in Geometry. It is beneficial to learn because of its relevance to other subjects. Furthermore, this concept is crucial from an academic standpoint because it is worth a lot of marks (28 marks from Unit 4, to be precise). As a result, learning this subject for exams is important, and one of the greatest resources is the RD Sharma Class 9 Solutions Chapter 11 - Coordinate Geometry (Ex 11.1) Exercise 11.1 - Free PDF. NCERT Solutions is also important due to its ease of use and diverse subject material. NCERT and RD Sharma Solutions are a few of the best study resources available easily. We made sure that the tough parts of the chapter were well-articulated and made easier to comprehend.
Download this PDF file to make your study sessions more fruitful and learn how to use the various concepts of coordinate geometry. Get to know how the expert mentors of Vedantu have formulated the solutions following the CBSE guidelines. become more confident by practicing solving these problems using the ways shown by the experts and gain more confidence.
The RD Sharma Class 9 Solutions Chapter 11 - Coordinate Geometry (Ex 11.1) Exercise 11.1 - Free PDF provides a comprehensive question-answer range for the chapter-Coordinate Geometry. RD Sharma offers different difficulty levels of questions. Students can get well trained in solving all mathematical problems with ease. You can practice the questions daily to enjoy the book's maximum benefits. The book should be studied regularly and students are required to practice daily.
The RD Sharma Class 9 Solutions Chapter 11 - Coordinate Geometry (Ex 11.1) Exercise 11.1 - Free PDF by Vedantu can help you with your examinations. The step-by-step explanations and the mentors' contributions at Vedantu help the student to learn mathematics at a deeper level. Once you are thorough with RD Sharma, you will be able to solve the most difficult problems at ease. The book offers an extensive range of questions. One can practice them daily and increase their level of understanding.
IIT- JEE is conducted on a national basis to select the most talented students for the field of engineering. The RD Sharma Class 9 Solutions Chapter 11 - Coordinate Geometry (Ex 11.1) Exercise 11.1 - Free PDF can benefit for the same. The syllabus of JEE is diverse and hence it is important to study every chapter of each grade in detail. Vedantu provides online coaching classes to help the students get prepared for JEE. To crack this exam, you need to prepare right from the Class 9 level and make your foundations stronger. As mathematics is a very scoring subject and its concepts are used in other science subjects, you will need this foundation in the long run.
No, currently RD Sharma Class 9 Solutions Chapter 11 - Coordinate Geometry (Ex 11.1) Exercise 11.1 - Free PDF is only available in English. However, our staff at Vedantu can teach you in your native language and help you with the guidance required. The book consists of simple English that is easy to understand. You can read and learn the subject. If you encounter any queries, our professional assistive team at Vedantu will be happy to help you.
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Vedantu is an online coaching and tutorial portal that assists students to learn and understand basic concepts of school education. Our team of experts is friendly and always ready to help. We provide step-by-step explanations for each topic and aim to deliver the best to the students. All the syllabus and guide books are provided in a free PDF format at Vedantu. Join us and learn better with online coaching classes. | 677.169 | 1 |
The Elements of Euclid
Dentro del libro
Resultados 6-10 de 36
Página 99 ... multiple of A , that L is of C ( 3. 5. ) ; for the same reason , M is the same multiple of B , that N is of D : and because , as A is to B , so is C to D ( Hypoth . ) and of A and C have been taken cer- tain equimultiples K , L ; and of ...
Página 100 ... multiple of A , that L is of C : and because A is to B , as C is to D , and of A and C certain equimultiples have ... multiple of another , which a magnitude taken from the first is of a magnitude taken from the other ; the remainder ...
Página 101 ... multiple of E , that KH is of F. But AB , by the hypothesis , is the same multiple of E that CD is of F ; therefore KH is the same multiple of F , that CD is of F ; wherefore KH is equal to CD ( 1. Ax . 5. ) : take away the common ...
Página 102 ... multiple of B that C is of D ; and that E is the same multiple of A that F is of C ; E is the same multiple of B that F is of D ( 3. 5. ) ; there- fore E and F are the same multiples of B and D : but G and H are equimultiples of B and D ...
Página 103 ... multiple , or part of the second ; the third is the same multiple , or the same part of the fourth . * Let A be to B , as C is to D ; and first let A be a multiple of B , C is the same multiple of D. Take E equal to A , and whatever | 677.169 | 1 |
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2016 AMC 10B Problems/Problem 14
Contents
Problem
How many squares whose sides are parallel to the axes and whose vertices have coordinates that are integers lie entirely within the region bounded by the line , the line and the line
Solution 1
The region is a right triangle which contains the following lattice points:
Squares :
Suppose that the top-right corner is , with . Then to include all other corners, we need .
This produces squares.
Squares :
Here . To include all other corners, we need .
This produces squares.
Squares :
Similarly, this produces squares.
No other squares will fit in the region. Therefore the answer is .
Solution 2
The vertical line is just to the right of , the horizontal line is just under , and the sloped line will always be above the value of .
This means they will always miss being on a coordinate with integer coordinates so you just have to count the number of squares to the left, above, and under these lines. After counting the number of , , and squares and getting , , and respectively, and we end up with .
Solution by Wwang
Solution 3
The endpoint lattice points are Now we split this problem into cases.
Case 1: Square has length .
The coordinates must be or and so on to The idea is that you start at and add at the endpoint, namely The number ends up being squares for this case.
Case 2: Square has length .
The coordinates must be or or and so now it starts at It ends up being | 677.169 | 1 |
The letter A labelled on the circle shows a _________.
plane
point
ray
space
Hint:
A point in geometry, refers to a dimensionless shape which does not have length, breadth or thickness. It just represents a dot.
The correct answer is: point
GIVEN- Given figure = circle Given letter = A TO FIND- What does letter A represent in the given figure. SOLUTION- From the given figure, we observe that letter A represents a dot inside the given circle. Hence, the letter A labelled on the circle shows a point. FINAL ANSWER- Option 'b' i.e. 'point' is the correct answer to the given question | 677.169 | 1 |
...AGH = alternate Z GHD, 3. and :. that AB |] CD. PROPOSITION XXIX. (Argument ad absurdum). Theorem. If a straight line fall upon two parallel straight lines, it makes the alternate angles equal to one another, and the exterior angle equal to the interior and opposite upon the same side; and theangle GHD ; and they are alternate angles ; therefore AB is parallel to CD. PROPOSITION XXIX. THEOREM. If a straight line fall upon two parallel straight lines, it makes the alternate angles equal to one another ; and the exterior angle equal to the interior and opposite upon the same side ; and...
...angles : therefore AB is parallel to CD. Wherefore, if a straight line, &c. QED PROP. XXIX. THEOR. IF a straight line fall upon two parallel straight lines, it makes the alternate angles equal to one another; and the exterior angle equal to the interior and opposite upon the same side ; and...
...geometrical, are nevertheless very ingenious and conclusive, see NOTES at the end. PROP. XXIX. THEOR. If a straight line fall upon two parallel straight lines, it makes the alternate angles equal to one another ; and the exterior angle equal to the interior and opposite upon the tame side ; and...
...alternate angles: therefore AB is parallel to CD. Wherefore, if a straight line, &c. QED PROP. XXIX. THEOR. If a straight line fall upon two parallel straight lines, it makes the alternate angles equal to one another; and the exterior angle equal to the interior and opposite upon the same side; and likewiseright angles ; AB is parallel to CD. But the 29th proposition assumes the converse of this, namely, if a straight line fall upon two parallel straight lines, it makes the alternate angles equal to one another ; and the exterior angle equal to the interior and opposite upon the same side ; and...
...alternate angles; therefore AB is parallel to CD. Wherefore if a straight line, &c. QED PEOP. XXIX. THEOR. IF a straight line fall upon two parallel straight lines it makes the alternate angles equal to one another ; and the exterior angle equal to the interior and opposite, upon the same side ; and...
...fore AB is parallelf to CD. Wherefore, if a straight line, &c. Q. i:. D. PROPOSITION XXIX. THEOR — If a straight line fall upon two parallel straight lines, it makes the alternate angles equal to one another ; and the exterior angle equal to the interior and opposite •upon the same side ;... | 677.169 | 1 |
Elements of Geometry
75. Two straight lines which are parallel to a third straight line are parallel to each other.
Let A B and CD be parallel to E F.
Since CD and E F are , HK is 1 to CD,
§ 67
(if a straight line be to one of two lis, it is to the other also).
Since AB and E F are , HK is also to A B,
$ 67
(when two straight lines are cut by a third straight line, if the ext.-int.
be equal, the two lines are ||).
Q. E. D.
PROPOSITION XVIII. THEOREM.
76. Two parallel lines are everywhere equally distant
Let A B and CD be two parallel lines, and from any two points in AB, as E and H, let EF and HK be drawn perpendicular to AB.
On MP as an axis, fold over the portion of the figure on the right of MP until it comes into the plane of the figure on the left.
(for
the point I will fall on E,
(for M HM E, by hyp.);
HK will fall on EF,
MHK = ≤ M E F, each being a rt. ▲);
and the point K will fall on E F, or E F produced.
Also, PD will fall on PC,
(ZMPK = ZMP F, each being a rt. 2);
and the point K will fall on PC.
Since the point K falls in both the lines EF and P C', it must fall at their point of intersection F.
77. Two angles whose sides are parallel, two and two, and lie in the same direction, or opposite directions, from their
Let&B and E (Fig. 1) have their sides B A and E D, and BC and E F respectively, parallel and lying in the same direction from their vertices.
We are to prove the LB = LE.
Produce (if necessary) two sides which are not || until they intersect, as at H;
Let B' and E' (Fig. 2) have B' A' and E' D', and B' C' and E' F' respectively, parallel and lying in opposite directions from their vertices.
We are to prove the LB'LE'.
Produce (if necessary) two sides which are not until they
78. If two angles have two sides parallel and lying in the same direction from their vertices, while the other two sides are parallel and lie in opposite directions, then the two angles are supplements of each other.
Let A B C and D E F be two angles having BC and E D parallel and lying in the same direction from their vertices, while E F and B A are parallel and lie in opposite directions.
We are to prove ABC and Z DEF supplements of each other.
Produce (if necessary) two sides which are not | until they intersect as at H.
But BHD and BHE are supplements of each other, § 34
(being sup.-adj. ▲).
ZABC and DEF, the equals of BHD and
ZBH E, are supplements of each other.
Q. E. D.
ON TRIANGLES.
79. DEF. A Triangle is a plane figure bounded by three straight lines.
A triangle has six parts, three sides and three angles.
80. When the six parts of one triangle are equal to the six parts of another triangle, each to each, the triangles are said to be equal in all respects.
81. DEF. In two equal triangles, the equal angles are called Homologous angles, and the equal sides are called Homologous sides.
82. In equal triangles the equal sides are opposite the equal angles.
SCALENE.
AA
ISOSCELES.
EQUILATERAL.
83. DEF. A Scalene triangle is one of which no two sides
are equal.
84. DEF. An Isosceles triangle is one of which two sides are equal.
85. DEF. An Equilateral triangle is one of which the three sides are equal.
86. DEF. The Base of a triangle is the side on which the triangle is supposed to stand.
In an isosceles triangle, the side which is not one of the equal sides is considered the base. | 677.169 | 1 |
Angle Of Rotation
Angle Of Rotation
The measure of the amount by which a figure is rotated about a point of rotation.
This measure is negative for a clockwise rotation and positive for a counterclockwise rotation.
An angle of rotation can be expressed using various measurement systems such as the sexagesimal system (in degrees, minutes and seconds) and the circular system (in radians) or by using numbers or fractions of turns. | 677.169 | 1 |
Course Content
Angle types
In this project the students work on constructing a mathematical model of the types of angles. The project is designed to be carried out in groups, allowing pupils to develop teamwork skills. The project also involves pupils using some of their geometry skills to draw details of the model. | 677.169 | 1 |
Transformation geometry
2. Measurement of the Earth.
In today's usage, it is a branch of
mathematics dealing with spatial
figures.
3. a process which changes the position
(and possibly the size and orientation)
of a shape. There are four types of
transformations: reflection, rotation,
translation and enlargement.
4. Historical Overview of Transformation
Geometry
17th century
Mathematician.
Made a great contribution
in analytic geometry.
First used the Cartesian
coordinate system.
Every point of a curve is
given two numbers that
represents its location in a
plane.
Rene Descartes
5. Historical Overview of Transformation
Geometry
Proposed a system of analytic
geometry similar to Descartes.
Credited because of his
independent developing ideas in
analytical geometry.
invented modern number
theory virtually single-handedly.
Formulated several theorems
on number theory, as well as
contributing some early work on
infinitesimal calculus.
6. Historical Overview of Transformation
Geometry
assign algebraic ideas to
geometric figures led to the
study of group theory in
geometry.
Enlanger Program
Study in geometry defined
as the study of transformations
that leave objects invariant.
Rearranged the unrelated
geometry know at his time
into a cohesive system.
Felix Klein ( 1849-1925)
7. Klein's Idea
• A geometry is a set of objects with the
rules determined by its symmetries, i.e., its
transformations. Two geometries may have
the same objects but different
transformations.
• The properties of the geometry are
properties that are not changed by the
transformations.
8. Different transformations
have been used in art,
architecture, crafts, and quilts
throughout history. Historians
have found numerous
transformation designs in
pottery, architecture, rugs,
quilts, and art pieces from
almost every culture. The
design used can help to
determine where and to
whom an artifact belonged.
9. Developed in the 7th, 8th,
9th centuries from beliefs
that creating a living
objects in art was
blasphemous or that God
should create animals and
other creatures in art work.
Due to this belief, many
did not use living creatures
in their art work, instead
they used different
transformations and
geometric designs to
increase the appeal of their
art and architecture.
Arabesque
10. Appliqué
a process when one
piece of fabric is sewn
onto another and then
stitched together with an
intricate design,
traditionally had
elaborate geometric
transformations that were
typically symmetric; such
as flowers and houses
that are not necessarily
symmetric.
11. Transformational geometry is quite important in
many fields, such as the study of architecture,
anthropology, and art, to name a few. The study of
which forms of transformations were used helps to
distinguish time frames for artifacts and helps to
illustrate which cultures may have made the item
being studied.
For example, architects are able to study the history
of very old buildings, taking note of which
transformations were used. A classical example that
involves this study is the illustration of the study of
the history of the Parthenon in Athens, Greece.
12. M. C. Escher (1898–1972)
Dutch graphic artist.
Escher would go to work on his pieces.
Escher read a few mathematics papers
regarding symmetry,
specifically George Pólya's (1887–1985)
1924 paper on 17 plane symmetry
groups, and although he
did not understand many of the ideas and
the mathematical theory of why it worked,
he did understand
the concepts of the paper and was able to
apply the ideas in his work. These
concepts helped him to use
mathematics more extensively throughout
many of his later pieces.
14. Reflection
You can reflect a figure using a line
or a point. All measures (lines and
angles) are preserved but in a mirror
image.
Example: The figure is reflected
across line l .
You could fold the picture along
line l and the left figure would
coincide with the corresponding
parts of right figure.
l
15. moves a shape by
sliding it up, down,
sideways or
diagonally, without
turning it or
making it bigger or
smaller.
Translation
16. Rotation
Rotation (also known as
Turn) turns a shape through a
clockwise or anti-clockwise angle
about a fixed point known as the
Centre of Rotation. All lines in the
shape rotate through the same
angle. Rotation, (just like
reflection) changes the orientation
and position of the shape, but
everything else stays the same.
17. Dilation
A dilation is a
transformation which
changes the size of a
figure but not its shape.
This is called a
similarity
transformation.
18. "Do not just pay attention
to the words;
Instead pay attention to
meaning behind the words.
But, do not just pay
attention to meanings
behind the words;
Instead pay attention to
your deep experience of
those meanings."
Tenzin Gyatso, The 14th
Dalai Lama
END | 677.169 | 1 |
Answer: C. ASA Congruence Postulate proves that △RMN and △QPN are congruent.Given: △RMN and △QPN where RN=NQ and ∠R=∠QTo prove: △RMN ≅△QPNProof:- In △RMN and △QPN∠R=∠Q............[given]RN=NQ............[given]∠RNM=∠PNQ............[vertically opposite angles]∴△RMN ≅△QPN [by ASA Congruence Postulate ]Therefore,C. is the right answer. ASA Congruence Postulate proves that △RMN and △QPN are congruent.ASA Congruence Postulate states that is two angles and the included side is equal to the two angles and the included side of the other triangle, then the triangles are said to be congruent. | 677.169 | 1 |
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GM5-9: Define and use transformations and describe the invariant properties of figures and objects under these transformations that students will accurately describe the transformation a figure or object has undergone and describe the invariant properties. At Level Five this will include the angle and centre of rotation, the distance and direction of translation, the magnitude and centre of enlargement (including fractional changes for example 1 1/2), and the line of reflection. Students will also draw the results of transformations on objects.
Students will be able to describe which properties of shapes change for each transformation:
Under rotation lengths, areas, angles do not change but orientation does.
Under reflection lengths, areas and angles do not change but orientation does.
Under translation lengths, areas, angles and orientation do not change.
Under positive enlargement angles and orientation do not change but lengths and areas do.
In this unit a number of small boxes are produced (by folding), that fit inside one another like Russian dolls. Lengths are measured and areas and volumes are calculated in order to make scale comparisons.
The purpose of this unit is to engage students in applying their understanding of geometric thinking to design and describe formations and the translations and/or rotations needed to create those formations. This is explored in the context of choreographing a synchronised swimming routine and...
Find the size of angles.
Describe the features of regular polygons, including the size of their interior angles. | 677.169 | 1 |
chapter outline
Octahedron
'Octa' means eight, and 'hedron' means base or seat. So, an octahedron is a 3-dimensional solid with eight flat faces.
Octahedron
Definition
An octahedron is a polyhedron. It has eight faces, twelve edges, and six vertices. If we join two square pyramids at their bases, we can create a regular octahedron. It belongs to the family of five platonic solids.
Octahedron-shaped diamonds, ornaments, dice, and Rubik's cubes are some real-life examples of an octahedron.
We can create the shape of a regular octahedron with the help of a 2-D octahedron net made from paper, as shown in the link below.
Octahedron Net
Source: mechamath.com
Properties
Has 8 faces. Each face is an equilateral triangle.
Has 12 edges.
Each face is an equilateral triangle.
Has 6 vertices. At each vertex, four edges meet.
Octahedron Faces Edges Vertices
The angles between the edges of an octahedron are 60° each, and the dihedral angle is approximately 109.47122° or 109.28°28′. | 677.169 | 1 |
12.1: The Faces of Geometry (5 minutes)
Warm-up
In this warm-up, students practice visualizing and drawing the faces of several solids. This will be helpful in upcoming activities as they categorize solids based on features of their choosing, and as they build solids from nets as a foundation for developing the formula for the volume of a pyramid.
Launch
Arrange students in groups of 2. Give students quiet work time and then time to share their work with a partner.
Student Facing
Student Response
Anticipated Misconceptions
Students may struggle to draw the cone surface that's in the shape of a sector of a circle. Ask them to consider snipping the cone in a straight line along this face and unrolling it.
Activity Synthesis
Here are questions for discussion.
"What are the names of these solids?" (rectangular pyramid, triangular prism, cone)
"What is the same and different about the surfaces of the prism and the pyramid?" (Each of these solids has 5 faces, and the faces are all triangles and rectangles. The prism has a triangle for a base and rectangles for the other faces, while the pyramid has a rectangle for a base and triangles for the other faces.)
"Which is the only surface that's not a polygon?" (The cone has one "face" shaped like a wedge from a circle.)
12.2: Card Sort: Sorting Shapes (10 minutes)
Activity
A sorting task gives students opportunities to analyze representations, statements, and structures closely and make connections (MP2, MP7). In this task, students sort solids based on features of their choosing. The structures students identify will allow them to extend the adjectives right and oblique to pyramids and cones.
Monitor for different ways groups choose to categorize the solids, but especially for categories that distinguish between right and oblique solids, and between solids that have an apex (pyramids and cones) and those that do not (prisms and cylinders).
As students work, encourage them to refine their descriptions of the solids using more precise language and mathematical terms (MP6).
Launch
Arrange students in groups of 2 and distribute pre-cut slips. Tell students that in this activity, they will sort some cards into categories of their choosing. When they sort the solids, they should work with their partner to come up with categories.
Conversing: MLR2 Collect and Display. As students work on this activity, listen for and collect the language students use to distinguish between right and oblique solids. Also, collect the language students use to distinguish between solids with an apex and solids with two congruent bases. Write the students' words and phrases on a visual display. As students review the visual display, create bridges between current student language and new terminology. For example, "the tip of the cone" is the apex of the cone. The phrase "the pyramid is slanted" can be rephrased as "the altitude of the pyramid does not pass through the center of the base." This will help students use the mathematical language necessary to precisely describe the differences between categories of solids. Design Principle(s): Optimize output (for comparison); Maximize meta-awareness
Student Facing
Your teacher will give you a set of cards that show geometric solids. Sort the cards into 2 categories of your choosing. Be prepared to explain the meaning of your categories. Then, sort the cards into 2 categories in a different way. Be prepared to explain the meaning of your new categories.
Student Response
Activity Synthesis
Select groups to share their categories and how they sorted their solids. Choose as many different types of categories as time allows, but ensure that one set of categories distinguishes between right and oblique solids, and another distinguishes between solids with an apex versus those without. Attend to the language that students use to describe their categories, giving them opportunities to describe their solids more precisely. Highlight the use of terms like hexagonal, perpendicular, and circular.
If students use phrasing such as: "The pyramids and cones get smaller while the cylinders and prisms do not," encourage them to use the language of cross sections. A sample response might be: "Cross sections taken parallel to the base of prisms and cylinders are congruent throughout the solid. On the other hand, cross sections taken parallel to pyramid and cone bases are similar to each other, but are not congruent."
Tell students that we can use the categories they created to define some characteristics of solids. A pyramid is a solid with one face (called the base) that's a polygon. All the other faces are triangles that all meet at a single vertex, called the apex. A cone also has a base and an apex, but its base is a circle and its other surface is curved.
Just like prisms and cylinders can be right and oblique, so can cones and some pyramids. For a cone, imagine dropping a line from the cone's apex straight down at a right angle to the base. If this line goes through the center of the base, then the cone is right. Otherwise, the cone is oblique. Pyramids with bases that have a center, like a square, a a pentagon, or an equilateral triangle, can also be considered right or oblique in the same way as cones.
For example, the cone on Card E from this activity is a right cone because its apex is centered directly over the center of its base. However, the pyramid on Card G has its center shifted; if we drop a height line straight down at right angles to the plane of the base, the line doesn't hit the center of the pyramid's base. This pyramid is oblique.
Point out that some mathematicians consider a cone to be a "circular pyramid," others consider pyramids to be "polygonal cones," and still others classify them in totally separate categories. Regardless of what we call them, the two kinds of solids share certain properties that will be explored in upcoming activities.
12.3: Building a Prism from Pyramids (15 minutes)
Activity
In this activity, students build a triangular prism out of 3 pyramids and make a conjecture about the volume of one of the pyramids. This activity creates a foundation for upcoming activities in which students will derive the formula for the volume of a pyramid.
Launch
Arrange students in groups of 3. Provide each group with scissors, tape, and 1 set of nets.
Tell students that they'll be building pyramids, and that they should save the pyramids when they're done for use in an upcoming activity.
Action and Expression: Internalize Executive Functions. Chunk this task into more manageable parts to support students who benefit from support with organization and problem solving. For example, present one step at a time and monitor students to ensure they are making progress throughout the activity. Supports accessibility for: Organization; Attention
Student Facing
Your teacher will give your group 3 nets. Each student should select 1 of the 3 nets.
Cut out your net and assemble a pyramid. The printed side of the net should face outward.
Assemble your group's 3 pyramids into a triangular prism. Each pair of triangles with a matching pattern will come together to form one of the rectangular faces of the prism. You will need to disassemble the prism in a later activity, so use only a small amount of tape (or no tape at all if possible).
Make a conjecture about the relationship between the volume of the pyramid marked P1 and the volume of the prism.
What information would you need to verify that your conjecture is true?
Don't throw away your pyramids! You'll use them again.
Student Response
Activity Synthesis
The goal of the discussion is to make observations about the relationships between the 3 pyramids and the prism. Here are some questions for discussion. It's okay for the answers about triangle congruence to be informal.
"Take a look at the pyramid marked P3. Which face would you consider its base? Is there only one possibility?" (Any face of this pyramid could be considered the base. No matter which face we choose to call the base, the remaining faces are all triangles. This is actually true for all 3 pyramids.)
"Which faces of the prism are its bases?" (The faces marked P1 and P3 are the prism's bases.)
"Do the pyramids marked P1 and P3 have any congruent faces? If so, which are they, and how do you know?" (The faces marked P1 and P3 are congruent because they are the prism's bases. The faces with the lines are also congruent, because together, they form a rectangle.)
"Do the pyramids marked P2 and P3 have any congruent faces? If so, which are they, and how do you know?" (The gray-colored faces are congruent because together, they form a rectangle. The two faces that are unmarked are congruent to each other. When assembled into the prism, each line segment that forms the sides of the triangles is shared between the two shapes.)
To ensure the pyramids are available for an upcoming activity, collect the assembled pyramids or direct students to place them in a safe storage area.
Speaking: MLR8 Discussion Supports. As students share the congruent faces between the pyramids marked P1, P2, and P3, press for details by asking how they know that the faces are congruent. Invite students to repeat their reasoning using mathematical language relevant to the lesson such as prism, base, and rectangle. For example, ask, "Can you say that again, using the term 'base'?" Consider inviting the remaining students to repeat these phrases to provide additional opportunities for all students to produce this language. This will help students justify their reasoning for why certain faces of the pyramids are congruent. Design Principle(s): Support sense-making; Optimize output (for justification)
Lesson Synthesis
Lesson Synthesis
The goal of the discussion is to consider what information would be needed to show that the volumes of the 3 pyramids are equal.
Display this image for all to see. Ask students, "How do the volumes of these 2 pyramids compare? How do you know?" Challenge them to use the language of cross sections. The bases of the pyramids have equal area, and the pyramids have the same height. Even though one pyramid's apex is shifted compared to the other pyramid, the cross sections of the two pyramids have the same area at all heights. Therefore, the pyramids have the same volume.
Description: <p>A pyramid with a square base. The sides of the base are labeled 5. The height, inside the pyramid, is labeled 7. Another pyramid, slanted to the right, with square base. The sides of the base are labeled 5. The height, outside the pyramid, is labeled 7.</p>
Choose 2 of the 3 assembled pyramids and display them for all to see. Ask students, "What would we need to know in order to verify the volumes of these two pyramids are equal?" We would need to know the pyramids have bases with equal area, and that the heights of the pyramids are equal. Then, the pyramids' cross sections would have equal area at all heights, and the pyramids would have equal volume.
12.4: Cool-down - How Many Faces? (5 minutes)
Cool-Down
Student Lesson Summary
Student Facing
Pyramids and cones are different from prisms and cylinders in that they have just one base and an apex, or a single point at which the other faces of the solid meet.
Cones are like cylinders and prisms in that they can be oblique or right. If a line dropped from the cone's apex at right angles to the base goes through the center of the base, then the cone is right. Otherwise, the cone is oblique. Pyramids that have a clear center in their bases can also be considered right or oblique.
We can use relationships between pyramids and prisms to build a formula for the volume of a pyramid. The image shows 3 square pyramids assembled into a cube. We'll use similar thinking, but with triangular pyramids and prisms, to create a pyramid volume formula in an upcoming lesson | 677.169 | 1 |
The synoptical Euclid; being the first four books of Euclid's Elements of geometry, with exercises, by S.A. Good
Dentro del libro
Pįgina 7 ... AC is equal to AB ; and because the point B is the centre of the circle ACE , 2. BC is equal to BA . But it has been proved that CA is equal to AB ; therefore CA , CB , are each of them equal to AB ; but things which are equal to the same ...
Pįįgina 10 ... equal to one another ; and if the equal sides be produced , the angles upon the other side of the base shall be equal . Let ABC be an isosceles triangle , of which the side AB is equal to 4C , and let the straight lines AB , AC , be ...
Pįgina 11 ... equal to one another , the sides also which subtend , or are opposite to , the equal angles , shall be equal to one another . Let ABC be a triangle having the angle ABC equal to the angle ACB ; the side AB is also equal to the side AC ...
Pįgina 12 ... ; much more then 3. The angle BDC is greater | 677.169 | 1 |
Proving Segment Relationships Worksheet
Proving Segment Relationships Worksheet - Web 5.2 i can prove segment and angle relationships given: Determining inscribed and central angles and determining pieces of secants and chords. Web the 9th row is changing segment length to segment congruence using definition of congruent segments. Web this is 6 worksheets on circles. A good proof is made of five essential parts: Web 5.3 proving segment relationships i can write proofs involving segment addition and congruence recall: Selection file type icon file name description size revision time user; Worksheets are 66580176, geometry chapter 2 reasoning and proof,. Web this is a set of 6 cut & paste segment relationship proofs. State the theorem to be proved.
Web 5.3 proving segment relationships i can write proofs involving segment addition and congruence recall: Web this starts out with algebraic proofs and then moves into proving statements about segments and angles, and finishes with. A good proof is made of five essential parts: State the theorem to be proved. Web this is 6 worksheets on circles. Worksheets are 66580176, geometry chapter 2 reasoning and proof,. Selection file type icon file name description size revision time user;
Proving segment relationships in this lesson: 2 versions of each proof are included for easy differentiation of. A good proof is made of five essential parts: Web the 9th row is changing segment length to segment congruence using definition of congruent segments. Worksheets are 66580176, geometry chapter 2 reasoning and proof,.
Proving Angle Relationships Worksheet Answers kidsworksheetfun
Web 5.3 proving segment relationships i can write proofs involving segment addition and congruence recall: Worksheets are 66580176, geometry chapter 2. Web students drag statements/reasons and drop them in the correct order of segment and angle addition proofs.included are 5 drag. Determining inscribed and central angles and determining pieces of secants and chords. In this lesson, we will learn how.
ShowMe Geometry 2.7 Proving segment relationships
Web open ended draw a representation of the segment addition postulate in which the segment is two inches long , contains four. Web this starts out with algebraic proofs and then moves into proving statements about segments and angles, and finishes with. Web this is a set of 6 cut & paste segment relationship proofs. Web included in this resource.
Honors Geometry Vintage High School Section 27 Proving Segment and
Web this is 6 worksheets on circles. Web 5.3 proving segment relationships i can write proofs involving segment addition and congruence recall: Web included in this resource is the (smart notebook) slide deck for geometry lesson 2.7 segment proofs. 2 versions of each proof are included for easy differentiation of. Web study with quizlet and memorize flashcards containing terms like.
B is the midpoint of ac prove: Web showing 8 worksheets for proving segment relationships. Worksheets are 66580176, geometry chapter 2. Worksheets are 66580176, geometry chapter 2 reasoning and proof,. In this lesson, we will learn how to write proofs involving addition of.
2.7 Notes Proving Segment Relationships YouTube
State the theorem to be proved. Web study with quizlet and memorize flashcards containing terms like segment addition postulate, segment addition postulate,. B is the midpoint of ac prove: Worksheets are 66580176, geometry chapter 2. Web the 9th row is changing segment length to segment congruence using definition of congruent segments.
Geometry Proving Segment Relationships YouTube
Web students drag statements/reasons and drop them in the correct order of segment and angle addition proofs.included are 5 drag. Web 5.2 i can prove segment and angle relationships given: 2 versions of each proof are included for easy differentiation of. In this lesson, we will learn how to write proofs involving addition of. State the theorem to be proved.
Proving Segment Relationships Worksheet - A good proof is made of five essential parts: Selection file type icon file name description size revision time user; Web this is 6 worksheets on circles. Determining inscribed and central angles and determining pieces of secants and chords. B is the midpoint of ac prove: 2 versions of each proof are included for easy differentiation of. Proving segment relationships in this lesson: Web 5.3 proving segment relationships i can write proofs involving segment addition and congruence recall: Web open ended draw a representation of the segment addition postulate in which the segment is two inches long , contains four. Web included in this resource is the (smart notebook) slide deck for geometry lesson 2.7 segment proofs.
Web study with quizlet and memorize flashcards containing terms like segment addition postulate, segment addition postulate,. Web this is a set of 6 cut & paste segment relationship proofs. Web included in this resource is the (smart notebook) slide deck for geometry lesson 2.7 segment proofs. 2 versions of each proof are included for easy differentiation of. Determining inscribed and central angles and determining pieces of secants and chords.
Web the 9th row is changing segment length to segment congruence using definition of congruent segments. State the theorem to be proved. Web included in this resource is the (smart notebook) slide deck for geometry lesson 2.7 segment proofs. A good proof is made of five essential parts:
Determining Inscribed And Central Angles And Determining Pieces Of Secants And Chords.
Worksheets are 66580176, geometry chapter 2. State the theorem to be proved. Web this starts out with algebraic proofs and then moves into proving statements about segments and angles, and finishes with. Web this is a set of 6 cut & paste segment relationship proofs.
Web Open Ended Draw A Representation Of The Segment Addition Postulate In Which The Segment Is Two Inches Long , Contains Four.
Web included in this resource is the (smart notebook) slide deck for geometry lesson 2.7 segment proofs. Web the 9th row is changing segment length to segment congruence using definition of congruent segments. A good proof is made of five essential parts: 2 versions of each proof are included for easy differentiation of. | 677.169 | 1 |
Types Of Angles Worksheet
Types of angles 1 free. Whether it is basic concepts like naming angles identifying the parts of an angle classifying angles measuring angles using a protractor or be it advanced like complementary and supplementary angles angles formed between intersecting lines or angles formed in 2d shapes we have them all covered for students.
The following table shows the different types of angles.
Types of angles worksheet. Acute obtuse right. Each worksheet has two problems for children to work on. Identifying types of indicated angles.
The angles worksheets are randomly created and will never repeat so you have an endless supply of quality angles worksheets to use in the classroom or at home. Each angles worksheet in this section also provide great practice for measuring angles with a protractor. Help students learn to differentiate between acute obtuse and right angles with these printables.
Right angles acute angles obtuse angles straight angles reflex angles and full angles. Types of angles worksheet is a very simple and quick resource to help check for the understanding of acute right obtuse and straight angles. This far from exhaustive list of angle worksheets is pivotal in math curriculum.
This page features many worksheets and a set of task cards. The actual angle is provided on the answer key so in addition to identifying whether an angle is obtuse or acute you can suggest that students mesaure the angle with a protractor and supply the measurement as part of their work. It will help to catch and redirect any misunderstandings before moving on to more in depth angle questions.
Observe the indicated angle and write the type of angle it exhibits. Examine your understanding of angle classification with these 6th grade worksheets wherein multiple angles share a common vertex. Scroll down the page if you need more explanations about each type of angles videos and worksheets.
The links below will connect you to sections of our site with geometry activities and printables on angles angle types and angle measurement. This page has printable geometry pdfs on angle types. Here is a graphic preview for all of the angles worksheets you can select different variables to customize these angles worksheets for your needs.
Most worksheets require students to identify or analyze acute obtuse and right angles. The more advanced worksheets include straight and reflex angles too. Geometry worksheets angles worksheets for practice and | 677.169 | 1 |
Show that the triangle
of maximum area that can be inscribed in a given circle is an equilateral
triangle.
Video Solution
Text Solution
Verified by Experts
Let 2r be the base and h be the heightof triangle, which is inscribed in a circle of radius R Area of triangle=12(base)(height) A=12(2r)(h)=rh…(1) Area being positive quantity, A will be maximum or minimum if A2 is maximum or minimum. Z=A2=r2h2…………(2) Now In triangle OLB BL2=OB2−OL2 In ΔOBD Z=A2=r2h2,r2=R2−(h−R)2⇒r2=2hR−h2put in (2) Z=h2(2hR−h2) ⇒Z=(2h3R−h4) ⇒dZdh=6h2R−4h3.......(3) For maxima/minimadZdh=0 ⇒6h2R−4h3=0 ⇒4R=4h(h≠0) ⇒h=3R2 differentiating (3) w.r.t . h ⇒d2Zdh2=12R−12h2 ⇒d2Zdh2]h=3R2=12(3R2)R−12(3R2)2 =182−27R2=−ve so, Z=A2 is maxima when h=3R2 ⇒A is maximum when h=3R2 when h=3R2,r2=2hR−h2=2R.3R2−(3R2)2 r2=3R24 r=√3R2 tanθ=hr=3R2√3R2=√3θ=π3 Triangle ABC is equilateral triangle | 677.169 | 1 |
// Define a function named pythagorean_theorem that calculates the hypotenuse of a right triangle given the lengths of its two legs.
function pythagorean_theorem(x, y) {
// Check if both x and y are of type number, if not, return false.
if ((typeof x !== 'number') || (typeof y !== 'number'))
return false;
// Calculate and return the square root of the sum of the squares of x and y (hypotenuse).
return Math.sqrt(x * x + y * y);
}
// Output the result of calculating the hypotenuse of a triangle with legs of lengths 2 and 4 to the console.
console.log(pythagorean_theorem(2, 4));
// Output the result of calculating the hypotenuse of a triangle with legs of lengths 3 and 4 to the console.
console.log(pythagorean_theorem(3, 4)); | 677.169 | 1 |
In a right-angled triangle ABC, angle B = 90 *, AB = 8cm, AC = 16cm.
In a right-angled triangle ABC, angle B = 90 *, AB = 8cm, AC = 16cm. Find the angles that form the height of the BH with the legs of the triangle.
1. Considering that, according to the properties of a right-angled triangle, the ratio of the leg AB to the hypotenuse AC is the sine of the angle at the vertex C, we calculate the value of this angle.
Sine of angle C = AB / AC = 8/16 = 1/2. Angle A = 30 °.
2. We calculate the value of the angle СBН between the height BН and the leg ВС of the triangle ABC:
angle СBН = 180 – 30 ° – 90 ° = 60 °.
3. We calculate the value of the angle ABH formed by the height BH and the leg AB of the triangle ABC:
angle ABH = 90 ° – 60 ° = 30 °.
Answer: angle СBН = 60 °, angle АBН = 180 ° – 60 ° = 30 | 677.169 | 1 |
I am taking the geometry approach. We know from intuition that more than three legs on a chair will make it unstable if any of the legs have a different length than the others. So by "wobble" I mean the possibility that at least one of the legs will be in the air when one or more legs are made shorter/longer than others. Also, the "surface" must be perfectly flat.
A three legged chair is unaffected by any amount of change we make to its legs. So to prove this I started out connecting lines between each legs (diagonals). So far I haven't made any progress.
For a triangle there are no diagonals. Is it enough to show that all the legs must be in the same plane for the chair to be stable?
1 Answer
1
If "wobble" means "one leg in the air", then "doesn't wobble" means "all legs touch the floor". For a mathematician, it should be enough that all legs lie in the same plane, since you can rotate the chair in such a way that this plane coincides with the plane of the floor.
From an egineering point of view, you might have additional constraints like the center of gravity should be above the convex hull of the points touching the floor and so on, but that has little to do with the number of legs, and only affects whether your chair is likely to fall over instead of wobble. | 677.169 | 1 |
The Diagonal of a Cube: Exploring its Properties and Applications
A cube is a three-dimensional geometric shape that is characterized by its six equal square faces, twelve edges, and eight vertices. One of the most intriguing aspects of a cube is its diagonal, which connects two opposite vertices of the cube. In this article, we will delve into the properties of the diagonal of a cube, its mathematical significance, and its practical applications in various fields.
Understanding the Diagonal of a Cube
The diagonal of a cube is a line segment that connects two opposite vertices of the cube, passing through its center. It is the longest possible line segment that can be drawn within the cube. The
For a cube with side length s, the length of the diagonal (d) can be calculated as:
d = √(s^2 + s^2 + s^2) = √(3s^2) = s√3
Therefore, the length of the diagonal of a cube is equal to the side length multiplied by the square root of three.
Properties of the Diagonal of a Cube
The diagonal of a cube possesses several interesting properties that make it a fascinating geometric concept. Let's explore some of these properties:
1. Length
As mentioned earlier, the length of the diagonal of a cube is equal to the side length multiplied by the square root of three. This property allows us to calculate the diagonal length of a cube if we know the side length, or vice versa.
2. Relationship with the Side Length
The diagonal of a cube is always longer than its side length. In fact, the diagonal is approximately 1.732 times longer than the side length. This relationship is derived from the Pythagorean theorem and is a fundamental property of cubes.
3. Relationship with the Face Diagonal
A cube also has face diagonals, which connect two opposite corners of a face. The length of a face diagonal is equal to the side length multiplied by the square root of two. Interestingly, the length of the diagonal of a cube is equal to the square root of three times the length of a face diagonal.
4. Relationship with the Space Diagonal
The space diagonal of a cube is the longest possible line segment that can be drawn within the cube, connecting two opposite corners. The length of the space diagonal is equal to the side length multiplied by the square root of three. Therefore, the diagonal of a cube is equal to the space diagonal divided by the square root of three.
Applications of the Diagonal of a Cube
The diagonal of a cube finds applications in various fields, ranging from mathematics to architecture and engineering. Let's explore some practical applications:
1. Volume and Surface Area Calculations
Knowing the length of the diagonal of a cube allows us to calculate its volume and surface area. The volume of a cube can be calculated using the formula V = s^3, where V represents the volume and s represents the side length. Similarly, the surface area of a cube can be calculated using the formula A = 6s^2, where A represents the surface area. By substituting the value of the diagonal length (d = s√3) into these formulas, we can obtain accurate volume and surface area calculations.
2. Structural Design and Analysis
In architecture and engineering, the diagonal of a cube plays a crucial role in structural design and analysis. Understanding the diagonal length helps architects and engineers determine the stability and load-bearing capacity of cube-shaped structures. By considering the diagonal length, they can make informed decisions regarding the placement of supports and the overall structural integrity of the design.
3. 3D Modeling and Computer Graphics
In the field of computer graphics and 3D modeling, the diagonal of a cube is used to create realistic and accurate representations of cube-shaped objects. By incorporating the correct diagonal length, designers can ensure that their virtual models accurately reflect the proportions and dimensions of real-life cubes.
4. Cubic Packing Efficiency
The diagonal of a cube is also relevant in the study of cubic packing efficiency. Cubic packing refers to the arrangement of cubes in a three-dimensional space, maximizing the use of available volume. The diagonal length of a cube is a crucial factor in determining the optimal packing arrangement and achieving the highest possible packing efficiency.
Summary
The diagonal of a cube is a fascinating geometric concept that connects two opposite vertices of the cube, passing through its center. It possesses several interesting properties, including its length, relationship with the side length, face diagonal, and space diagonal. The diagonal of a cube finds applications in various fields, such as volume and surface area calculations, structural design and analysis, 3D modeling and computer graphics, and cubic packing efficiency. Understanding the properties and applications of the diagonal of a cube allows us to appreciate its mathematical significance and its practical relevance in different domains.
Q&A
1. What is the formula for calculating the length of the diagonal of a cube?
The length of the diagonal of a cube can be calculated using the formula d = s√3, where d represents the diagonal length and s represents the side length of the cube.
2. How does the length of the diagonal of a cube relate to its side length?
The length of the diagonal of a cube is approximately 1.732 times longer than its side length. This relationship is derived from the Pythagorean theorem and is a fundamental property of cubes.
3. What is the relationship between the diagonal of a cube and its face diagonal?
The length of the diagonal of a cube is equal to the square root of three times the length of a face diagonal. In other words, the diagonal is √3 times longer than the face diagonal.
4. How is the diagonal of a cube used in structural design and analysis?
The diagonal of a cube helps architects and engineers determine the stability and load-bearing capacity of cube-shaped structures. By considering the diagonal length, they can make informed decisions regarding the placement of supports and the overall structural integrity of the | 677.169 | 1 |
Periscope works on the principle of multiple reflection. It consists of two or more mirror, used for the reflection. If the two mirrors are placed at an angle of 90∘, how many images of the object are formed between them?
A
One
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B
Two
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C
Three
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D
Four
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Open in App
Solution
The correct option is C
Three
We use the formua n=(360∘Θ)−1. Which gives (360∘90∘)−1 = 3 Hence three images are formed. | 677.169 | 1 |
Quadrilaterals are grouped by their angles by the lengths of their sides and by any parallel sides.
Classifying quadrilaterals worksheet pdf. Harcourt rw94 reteach name lesson 18 3 classify quadrilaterals a quadrilateral has four sides and four angles. Preview images of the first and second if there is one pages are shown. I do not have any square corners.
Quadrilateral trapezoid square parallelogram rectangle rhombus kite find the. If there are more versions of this worksheet the other versions will be available below the preview images. 362 esson 32 classify quadrilaterals urriculum ssociates opying is not permitted use the clues and shapes a e to solve problems 6 8.
Add to my workbooks 1 download file pdf embed in my website or blog add to google classroom. I have all square corners. Use the buttons below to print open or download the pdf version of the classifying quadrilaterals a math worksheet. | 677.169 | 1 |
Ex 3.1 Class 9 Maths Question 1. How will
you describe the position of a table lamp on your study table to another
person?
Solution: To describe
the position of a table lamp placed on the table, let us consider the table
lamp as P and the table as a plane. Now, take
two mutually perpendicular edges of the table as the axes OX and OY. Measure the
perpendicular distance 'a' cm of P (lamp) from OY and the perpendicular
distance 'b' cm of P (lamp) from OX. Thus, the
position of the table lamp P is described by the ordered pair (a, b).
Let us assume that the distance of lamp from OY is 18 cm and
the distance of lamp from OX is 16 cm. Then the position of the lamp can be
described as the ordered pair (18, 16).
Ex 3.1 Class 9 Maths Question 2. (Street
Plan): A city has two main roads which cross each other at the centre of the
city. These two roads are along the North-South direction and East-West
direction. All other streets of the city run parallel to these roads and are
200 m apart. There are 5 streets in each direction. Using 1 cm = 200 m, draw a
model of the city on your notebook. Represent the roads/streets by single
lines. There are
many cross-streets in your model. A particular cross-street is made by two
streets, one running in the North-South direction and another in the East-West
direction. Each cross street is referred to in the following manner: If the 2nd street
running in the North-South direction and 5th in the East-West
direction meet at some crossing, then we will call this cross-street (2, 5).
Using this convention, find: (i) how many
cross-streets can be referred to as (4, 3). (ii) how many
cross-streets can be referred to as (3, 4).
Solution:
Let us draw two mutually perpendicular lines as the
two main roads of the city that cross each other at the centre and let us take
it as N-S and E-W direction.
Let us use the scale as 1 cm = 200 m.
Now, draw five streets that are parallel to both the
main roads, to get the given below figure.
(i) From the
figure, we can see that only one point has the coordinates as (4, 3). Thus, we
can conclude that only one cross street can be referred to as A(4, 3). (ii) From the figure,
we can see that only one point has the coordinates as (3, 4). Thus, we can
conclude that only one cross street can be referred to as B(3, 4). | 677.169 | 1 |
GoGeometry Action 170!
Creation of this resource was inspired by a problem posted by Antonio Gutierrez.
You can move the LARGE WHITE POINTS anywhere you'd like at any time.
Key Questions:
1) What is the EXACT MEASURE of the PINK ANGLE?
2) Suppose we didn't see the green triangle appear. How can we prove that this pink angle has this given
measure? | 677.169 | 1 |
JAMB CBT PRACTICE(mathematics)
Each of the interior angles of a regular polygon is 140°. How many sides has the polygon?
a. 5
b. 7
c. 8
d. 9
2 / 40
2) Calculate the median of 14, 17, 10, 13, 18 and 10.
a. 12.5
b. 13.5
c. 14.5
d. 11.5
3 / 40
3)
The difference between an exterior angle of (n - 1) sided regular polygon and an exterior angle of (n + 2) sided regular polygon is 6o, then the value of "n" is
a. 11
b. 13
c. 12
d. 14
4 / 40
4) List all integers satisfying the inequality in -2 < 2x-6 < 4
a. 2,3,4 and 5
b. 2,3
c. 4,5
d. 3,4
5 / 40
5) If n(P) = 20 and n(Q) = 30 and n(PuQ) = 40, find the value n(PnQ).
a. 10
b. 30
c. 15
d. 40
6 / 40
6)
A boat sails 8 km north from P to Q and then sails 6 km west from Q to R. Calculate the bearing of R from P. Give your answer to the nearest degree.
a. 323 degrees
b. 217 degrees
c. 037 degrees
d. 057 degrees
7 / 40
7)
Three times a certain number (x), minus 2 is less than the number minus 6. Find the possible values of x.
a. X>2
b. X<2
c. X<-2
d. X>-2
8 / 40
8) Find the compound interest (CI) on ₦15,700 for 2 years at 8% per annum compounded annually.
a. ₦6,212.48
b. ₦2,834.48
c. #18,312.48
d. #2612.48
9 / 40
9)
The locus of points equidistant from a fixed point is a___
a. Circle
b. Bisector
c. Straight lines
d. Perpendicular line
10 / 40
10) If −2x3+6x2+17x - 21 is divided by (x+1), then the remainder is
a. 30
b. 32
c. -32
d. -30
11 / 40
11) Find x if the mean of 2x, 4x, 2x - 13 and 6x is 4.
a. 2.0
b. 1.5
c. 0.5
d. 1.0
12 / 40
12)
Solve the logarithmic equation: log2(6−x)=3−log2x
a. X= 4 or 2
b. X= -4 or -2
c. X= -4 or 2
d. X= 4 or -2
13 / 40
13) A student is using a graduated cylinder to measure the volume of water and reports a reading of 18 mL. The teacher reports the value as 18.4 mL. What is the student\'s percent error?
a. 2.76%
b. 1.73%
c. 1.97%
d. 2.17%
14 / 40
14)
An article when sold for ₦230.00 makes a 15% profit. Find the profit or loss % if it was sold for ₦180.00
a. 10% gain
b. 12% loss
c. 12% gain
d. 10% gain
15 / 40
15) Find the probability that a number selected at random from 41 to 56 is a multiple of 9
a. 1/8
b. 7/8
c. 2/15
d. 3/16
16 / 40
16) A man sells different brands of an items. 1/9 of the items he has in his shop are from Brand A, 5/8 of the remainder are from Brand B and the rest are from Brand C. If the total number of Brand C items in the man\'s shop is 81, how many more Brand B items than Brand C does the shop has?
a. 243
b. 108
c. 54
d. 135
17 / 40
17) Two dice are tossed. What is the probability that the total score is a prime number.
a. 5/12
b. 5/9
c. 1/6
d. 1/3
18 / 40
18)
A group of market women sell at least one of yam, plantain and maize. 12 of them sell maize, 10 sell yam and 14 sell plantain. 5 sell plantain and maize, 4 sell yam and maize, 2 sell yam and plantain only while 3 sell all the three items. How many women are in the group?
a. 24
b. 17
c. 19
d. 25
19 / 40
19)
In how many ways can the letter of ZOOLOGY be arranged?
a. 720
b. 360
c. 120
d. 840
20 / 40
20)
If y varies inversely as x and x = 3 when y =4. Find the value of x when y = 12.
a. 4
b. 1
c. 2
d. 3
21 / 40
21)
Given that the cost C of running a school is directly proportional to the number of students N, if 20 students cost ₦10,000, How many students can ₦1,000,000 cover?
a. 3000
b. 1000
c. 2000
d. 4000
22 / 40
22)
Tanθ is positive and Sinθ is negative. In which quadrant does θ lies
a. Third only
b. Fourth only
c. First and Second only
d. Second only
23 / 40
23) Find the rate of change of volume V of a hemisphere with respect to its radius r, when r = 2
a. 8
b. 2
c. 16
d. 4
24 / 40
24) If the binary operation ∗ is defined by m ∗ n = mn + m + n for any real number m and n, find the identity of the elements under this operation
a. e=1
b. e=-1
c. e=-2
d. e=0
25 / 40
25) 4, 16, 30, 20, 10, 14 and 26 are represented on a pie chart. Find the sum of the angles of the bisectors representing all numbers equals to or greater than 16
a. 84 degrees
b. 48 degrees
c. 96 degrees
d. 276 degrees
26 / 40
26) Three children shared a basket of mangoes in such a way that the first child took 1/4 of the mangoes and the second 3/4 of the remainder. What fraction of the mangoes did the third child take?
a. 3/16
b. 7/16
c. 9/16
d. 13/16
27 / 40
27) What is the general term of the sequence 3, 8, 13, 18, ...?
a. 5n
b. 2
c. 5n + 2
d. 5n - 2
28 / 40
28) The third term of an A.P is 6 and the fifth term is 12. Find the sum of its first twelve terms
a. 201
b. 144
c. 198
d. 72
29 / 40
29) If sin θ = -3/5 and θ lies in the third quadrant, find cos θ
a. -4/5
b. 4/5
c. 5/4
d. -5/4
30 / 40
30)
Find the value of p if the line which passes through (-1, -p) and (-2,2) is parallel to the line 2y+8x-17 = 0
a. 2/7
b. -7/2
c. 7
d. 2
31 / 40
31)
Two numbers are respectively 35% and 80% more than a third number. The ratio of the two numbers is
a. 7:16
b. 3:4
c. 16:7
d. 4:3
32 / 40
32)
P(-6, 1) and Q(6, 6) are the two ends of the diameter of a given circle. Calculate the radius.
a. 6.5units
b. 13units
c. 3.5units
d. 7units
33 / 40
33)
Give that X is due east point of Y on a coast. Z is another point on the coast but 6.0km due south of Y. If the distance ZX is 12km, calculate the bearing of Z from X
a. 240 degrees
b. 150 degrees
c. 60 degrees
d. 270 degrees
34 / 40
34)
How many different 8 letter words are possible using the letters of the word SYLLABUS?
a. (8-2)!
b. 8!
c. 8/2!
d. 8/2!2!
35 / 40
35)
Find the length of a chord 3cm from the centre of a circle of radius 5cm.
a. 2cm
b. 8cm
c. 5.6cm
d. 1.6cm
36 / 40
36) Find the length of a side of a rhombus whose diagonals are 6cm and 8cm
a. 8cm
b. 5cm
c. 4cm
d. 3cm
37 / 40
37)
Find the equation of a straight line parallel to the line 2x - y = 5 and having intercept of 5
a. 2x-y=-5
b. 2x+y=-5
c. 2x+y=5
d. 2x-y=5
38 / 40
38) Bello buys an old bicycle for ₦9,200.00 and spends ₦1,500.00 on its repairs. If he sells the bicycle for ₦13,400.00, his gain percent is
a. 25.23%
b. 31.34%
c. 88.81%
d. 42.23%
39 / 40
39) If a fair coin is tossed twice, what is the probability of obtaining at least one head?
a. 0.50
b. 0.35
c. 0.75
d. 0.25
40 / 40
40)
Give that the mean of 2, 5, (x+1), (x+2), 7 and 9 is 6, find the median. | 677.169 | 1 |
Measuring Angles Worksheet With Protractor
Measuring Angles Worksheet With Protractor - Try to measure the angles a, b and c inside the triangle. Inner and outer scales of a protractor. Use these handy worksheets to teach your students how to use a. The protractor worksheets and blank printable projectors on this page require students to measure angles and identify whether they are right,. Web the measuring angles with a protractor worksheets will help the students learn about these different types of angles. Line up one side of the angle with the zero line of the. His or her job is to use a standard protractor to measure the angles. Web these measuring angles task cards can be used for math centers, guided math, independent practice, morning work, or. Use this resource when teaching students how to measure. Web a set of 3 worksheets to practice measuring angles using a protractor.
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Measuring Angles using a protractor activity
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Measuring Angles With A Protractor Worksheet —
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Measuring Angles With A Protractor Worksheet —
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Reading a Protractor Angles Worksheets on Naming Angles Worksheet
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Web How Do I Teach Measuring Angles Using A Protractor?
The protractor worksheets and blank printable projectors on this page require students to measure angles and identify whether they are right,. Web these printable geometry worksheets will help students learn to measure angles with a protractor, and draw angles with a given measurement. Use these handy worksheets to teach your students how to use a. Web a protractor is a transparent plastic tool used to measure angles from 0° to 180°—the two sets of numbers on a protractor increase in opposite directions.
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Web welcome to the angles, lines, and polygons worksheets section at tutorialspoint.com.on this page, you will find worksheets on. Web a set of 3 worksheets to practice measuring angles using a protractor. Try to measure the angles a, b and c inside the triangle. Line up one side of the angle with the zero line of the. | 677.169 | 1 |
dip angle वाक्य
These bodies generally strike ESE to SSE ( N 120 to N 150 ) in the northern branch, with a medium dip angle between 30 and 60?
trigonometric equation relating the dip angle ( between the true horizon and astronomical horizon ) observed from the top of a mountain to the height of that mountain . ]]
He applied the values he obtained for the dip angle and the mountain's height to the following trigonometric formula in order to calculate the Earth's radius:
One technique is to always take the strike so the dip is 90?to the right of the strike, in which case the redundant letter following the dip angle is omitted ( right hand rule, or RHR ).
Because the slope of a homoclinal ridge dips in the same direction as the sedimentary strata underlying it, the dip angle of this bedding ( ? ) can be calculated by v / h = tan ( ? ) where v is equal to the vertical distance and h is equal to the horizontal distance perpendicular to the strike of the beds. | 677.169 | 1 |
Is a Rectangle a Special Quadrilateral: Exploring the Unique Characteristics
Is a Rectangle a Special Quadrilateral: Exploring the Unique Characteristics
A rectangle is indeed a special type of quadrilateral, possessing unique characteristics that set it apart from other polygons. In this article, we will delve into the defining features of a rectangle and understand why it holds a special place in the world of quadrilaterals.
Defining a Rectangle
A rectangle is a four-sided polygon with opposite sides that are equal in length and four right angles. The presence of right angles makes it a member of the larger family of quadrilaterals, which encompasses various other shapes such as squares, parallelograms, and trapezoids.
Unique Characteristics
1. Equal Opposite Sides: One of the key characteristics of a rectangle is that its opposite sides are equal in length. This property ensures that the shape remains balanced and symmetrical.
2. Right Angles: Unlike other quadrilaterals, a rectangle has four right angles, each measuring 90 degrees. This property gives it a distinct geometric structure and makes it suitable for various practical applications.
3. Diagonals: The diagonals of a rectangle are congruent and bisect each other. This means that the line segments connecting opposite corners of the rectangle are of equal length and intersect at their midpoint. This property further enhances the symmetry of the shape.
4. Area and Perimeter: The area of a rectangle can be calculated by multiplying the length and width of the shape. Additionally, the perimeter of a rectangle is obtained by adding the lengths of all four sides. These formulas make it easier to determine the size and dimensions of a rectangle.
Applications of Rectangles
The unique characteristics of rectangles make them highly versatile and applicable in various fields. Some common applications include:
1. Architecture and Construction: Rectangles are commonly used in architectural designs and construction plans. The right angles and equal sides make them ideal for creating stable structures, such as buildings and bridges.
2. Mathematics and Geometry: Rectangles play a crucial role in geometry and mathematical calculations. They serve as the foundation for understanding concepts like area, perimeter, and coordinate geometry.
3. Graphic Design and Art: The symmetrical nature of rectangles makes them aesthetically pleasing and visually appealing. They are often used as frames, borders, and canvases in graphic design and art compositions.
4. Furniture and Interior Design: Many furniture pieces, such as tables, desks, and cabinets, are rectangular in shape. The balanced proportions and right angles of rectangles make them practical choices for designing functional and visually pleasing furniture.
In conclusion, a rectangle is indeed a special quadrilateral due to its unique characteristics. The presence of equal opposite sides, right angles, congruent diagonals, and formulas for area and perimeter make it a versatile shape with numerous applications in various fields. Understanding the distinct features of a rectangle helps us appreciate its significance in geometry, design, and everyday life | 677.169 | 1 |
Triangles
The word triangle is made from two words – "tri" which means three and "angle". Hence, a triangle can be defined as a closed figure that has three vertices, three sides, and three angles. The following figure illustrates a triangle ABC –
Vertices of a Triangle
In the above triangles, the three vertices are A, B and C.
Angles of a Triangle
The three angles are the angles made at these vertices, i.e. ∠A, ∠B and ∠C. The angle formed at A can also be written as ∠BAC. Similarly, we can write ∠ABC and ∠ACB. These angles are also called the interior angles of a triangle. An exterior angle of a triangle is formed by any side of a triangle and the extension of its adjacent side.
Sides of a Triangle
The three sides of the triangle above are AB, BC and AC.
What is an Isosceles Triangle?
The term isosceles triangle is derived from the Latin word 'īsoscelēs', and the ancient Greek word 'ἰσοσκελής (isoskelḗs)' which means "equal-legged". Also, ancient Babylonian and Egyptian mathematicians were of the know-how on the calculations required to find the 'area' much before the ancient Greek mathematicians started studying the isosceles triangle.
Now, let us understand the definition of an isosceles triangle.
A triangle is said to be an Isosceles triangle if its two sides are equal. If two sides are equal, then the angles opposite to these sides are also equal.
For example, in the following triangle, AB = AC. Therefore ∆ABC is an Isosceles triangle.
Since AB = AC
∠B = ∠C
Based on the interior angles, the isosceles triangle can further be divided into the following three types –
Right Isosceles Triangle
A triangle is said to be a right isosceles triangle if apart from two sides being equal, one of the angles of the triangle is a right angle, i.e. 90o. Suppose, we have a triangle, ABC where AB = BC and ∠ABC = 90o. Then such a triangle is called a right isosceles triangle which would be of a shape similar to the below figure.
Acute Isosceles Triangle
A triangle is said to be an acute isosceles triangle if apart from two sides being equal, all the three interior angles of the triangle are acute angles, i.e. all the three angles are less than 90o. Suppose, we have a triangle, ABC where AB = BC and all the three angles are acute angles. Then such a triangle is called an acute isosceles triangle which would be of a shape similar to the below figure.
Obtuse Isosceles Triangle
A triangle is said to be an obtuse isosceles triangle if apart from two sides being equal, one of the angles of the triangle is an obtuse angle, i.e. greater than 90o. Suppose, we have a triangle, ABC where AB = BC and ∠ABC > 90o. Then such a triangle is called an obtuse isosceles triangle which would be of a shape similar to the below figure.
Properties of Isosceles Triangles
Properties of Isosceles triangles can be stated as under –
Sides, Angles and Vertices
We already know that an Isosceles triangle always has exactly three sides and three vertices. This is the basic property of any triangle.
Angle Sum Property of an Isosceles Triangle
The sum of the measure of the three interior angles of an Isosceles triangle is always 180o.
In the above triangle, ∠x + ∠y + ∠z = 180o
Triangle Inequality Property
According to the triangle inequality property, the sum of two sides of an Isosceles triangle is always greater than or equal to the third side. If this sum is less than the third side, it is not possible to construct the triangle. For example, in a triangle ABC,
If one of the angles of a triangle is 90o, the sides that make the right angle are called the base and the perpendicular while the third side is called the hypotenuse.
According to Pythagoras Theorem
In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides.
Mathematically,
Base2 + Perpendicular2 = Hypotenuse2
Therefore, if "a" is the base, "b" is the perpendicular and "c" is the hypotenuse in a right angled triangle, then
c2 = a2 + b2
Exterior Angle Property
According to this property, the exterior angleof an Isosceles triangle is always equal to the sum of the interior opposite angles.
For example, in the above triangle, the exterior angle a equals the sum of the interior angles b and c.
∠a = ∠b + ∠c
Area of an Isosceles Triangle
Before we learn how to find the area of an isosceles triangle, we must first understand what we mean by the area of a triangle?
We must recall that a triangle is a polygon that is made of three edges and three vertices. The vertices join together to make three sides of a triangle. The area occupied between these three sides is called the area of a triangle.
In general, the area of an Isosceles triangle is defined by $\frac{1}{2} x b x h$
The following figure illustrates the base and the height of a triangle
The area of an isosceles triangle can be calculated using the general formula as stated above. But, can we calculate the area of an isosceles triangle if we know the two equal sides and the base? Let us find out.
Area of an Isosceles Triangle when the two Equal Sides and the Base is known
Let ABC be an isosceles triangle such that AB = AC = b units and BC = a units
Hence, the area of an isosceles triangle = $\frac{1}{2}\:x\:Base\:x\: \sqrt{(Equal\:Side)^2 – \frac{1}{4}((Base)^2}$
Now, let us learn how to find area of a right angled isosceles triangle.
Area of a Right angled Isosceles Triangle
From the definition of a right angled isosceles triangle that we have learnt above, it can be seen that a right angled isosceles triangle has a defined altitude in the form of the side AB and the side BC is its base. Hence the area of a right angled isosceles triangle can be calculated using the general formula only, which is given by
Area of a triangle = $\frac{1}{2}$ x b x h
where b = base of the triangle, and h = Height of the triangle from that base (or side)
Let us now learn about the Isosceles Triangle theorem that is related to finding the area of an isosceles triangle.
Isosceles Triangle TheoremThis further implies that ∠ B = ∠ C [ This is due to the rule that Corresponding parts of congruent triangles are equal ]
Now, that we have learnt how to find the area of an isosceles triangle, let us see how to find its perimeter.
Perimeter of a Triangle
The sum of the lengths of the sides of the isosceles triangle is called its perimeter. If the lengths of the sides of an isosceles triangle are a, a and b units, then,
Perimeter of the Isosceles Triangle = a + a + b = 2a + b units.
Let us understand this by an example.
Suppose we want to find the perimeter of the following triangle –
In the above figure, we can clearly see that two of the sides of the triangle are equal. This means that the given triangle is an isosceles triangle. Now the perimeter of the triangle is given by the sum of all its sides. Hence,
Perimeter of the given triangle = 5.6 cm + 5.6 cm + 2.8 cm = 14 cm.
Hence, the perimeter of the given triangle = 14 cm
Let us now learn about some other theorems related to isosceles triangles.
Theorems of Isosceles Triangles
The Isosceles Decomposition Theorem – In an isosceles triangle, if a line segment goes from the vertex angle to the base, the following conditions are equivalent:
The line segment meets the base at its midpoint,
The line segment is perpendicular to the base.
The line segment bisects the vertex angle.
If the bisector of an angle in a triangle is perpendicular to the opposite side, the triangle is isosceles.
If the line from an angle of a triangle that is perpendicular to the opposite side meets the opposite side at its midpoint, then the triangle is isosceles.
If the bisector of an angle in a triangle meets the opposite side at its midpoint, then the triangle is isosceles.
A point is on the perpendicular bisector of a line segment if and only if it lies the same distance from the two endpoints.
Solved Examples
Example 1 One of the angles of a triangle is 80o and the other two angles are equal. Find these angles.
Solution Since the other two angles are equal, let each of these angles be x. By angle sum property, the sum of the measure of the three interior angles of a triangle is always 180o.
Therefore,
x +x + 80o = 180o
2x +80o = 180o
2x = 100o
x = 50o
The angles each measure 50o
Example 2 Find the value of x in the following figure –
Solution We have been given an isosceles triangle, where x y = x z.
Now, we know that by the angle sum property of a triangle, the sum of three angles of a triangle is always equal to 180o.
Therefore, we can say that
∠ x + ∠ y + ∠ z = 180o
Now, since xy = xz, therefore, by isosceles traingel tehorem,
∠ y = ∠ z = xo
Also, we have been given that ∠ x = 84o
Therefore, we have,
84 + x + x = 180o
84 + 2x = 180o
2x = 180 – 84
2x = 96
x = 48
Hence, the value of x = 48o
Example 3Find x° and y° from the given figure –
Solution We have been given a triangle XYP. Now in this triangle, we can see that a linear pair is formed by ∠YXP and ∠QXY. Since the sum of a linear pair is 180° therefore,
∠YXP + ∠QXY = 180°
⇒ ∠YXP = 180° – ∠QXY
⇒ ∠YXP = 180° – 130°
⟹ ∠YXP = 50°
Now, that we have found the value of ∠YXP, let us now find the value of ∠XPY.
We have been given that XP = YP
Therefore,
⟹ ∠YXP = ∠XYP = 50°.
Also, by the angle sum property of a triangle, we know that the sum of the three angles of a triangle is always equal to 180o. Therefore,
∠YXP + ∠XYP + ∠XPY = 180°
⟹ 50° + 50° + ∠XPY = 180°
⟹ 100o + ∠XPY = 180°
⟹ ∠XPY = 180° – 100o
⟹ ∠XPY = 80°
Again, we can see that x° and ∠XPY form a linear pair. Therefore, we can say that,
x° + ∠XPY = 180°
⟹ x° + 80° = = 180°
⟹ x° = 180° – 80°
⟹ x° = 100°
Also, in ∆XPZ we have,
XP = ZP
Therefore, ∠PXZ = ∠XZP = z°
Therefore, in ∆XPZ we have,
∠XPZ + ∠PXZ + ∠XZP = 180°
⟹ x° + z° + z° = 180°
⟹ 100° + z° + z° = 180°
⟹ 100° + 2z° = 180°
⟹ 2z° = 180° – 100°
⟹ 2z° = 80°
⟹ z° = 80°280°2
⟹ z° = 40°
Therefore, y° = ∠XZR = 180° – ∠XZP
⟹ y° = 180° – 40°
⟹ y° = 140°
Example 4Find the area of an isosceles triangle given its height as 8 cm and base as 6 cm?
SolutionWe have been given that the height of the triangle is 8 cm while the base of the triangle is 6 cm.
We are required to find the area of the isosceles triangle. We know that if we have the base and the height of a triangle, its area is given by
Area of a Triangle = ½ × b × h
Therefore, substituting the values of the base and the height in the above formula, we have,
Area of a Triangle = ½ × 6 × 8 = 24
Hence, area of the given isosceles triangle = 24 sq. cm
Key Facts and Summary
A triangle is said to be an Isosceles triangle if its two sides are equal. If two sides are equal, then the angles opposite to these sides are also equal.
A triangle is said to be a right isosceles triangle if apart from two sides being equal, one of the angles of the triangle is a right angle, i.e. 90o.
A triangle is said to be an acute isosceles triangle if apart from two sides being equal, all the three interior angles of the triangle are acute angles, i.e. all the three angles are less than 90o.
A triangle is said to be an obtuse isosceles triangle if apart from two sides being equal, one of the angles of the triangle is an obtuse angle, i.e. greater than 90o.
The area occupied between these three sides is called the area of a triangle.
In general, the area of an Isosceles triangle is defined by ½ x b x h. This means that Area of a triangle = ½ x b x h, where b = base of the triangle (or any one side of the triangle), and h = Height of the triangle from that base (or side).
the area of an isosceles triangle if we know the two equal sides and the base is given by ½ x Base x $\sqrt{(Equal\:Side)^2- \frac{1}{4}(Base)^2}$The sum of the lengths of the sides of the isosceles triangle is called its perimeter.
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Tag: 1.3 midpoint and distance formula worksheet
Midpoint And Distance Formula Worksheet. Allowed in order to my blog, on this moment I'm going to teach you in relation to Midpoint And Distance Formula Worksheet. Why don't you consider graphic earlier mentioned? is usually which amazing???. if you feel consequently, I'l m demonstrate some picture once again beneath: So, if you'd like to… | 677.169 | 1 |
Parallel Lines Transversals Mrs Wedgwood Geometry Parallel Lines
Parallel Lines and Transversals What would you call two lines which do not intersect? _______ Exterior Interior Exterior A solid arrow placed on two lines of a diagram indicate the lines are parallel. The symbol ____ is used to indicate parallel lines. AB ___ CD
Parallel Lines and Transversals A slash through the parallel symbol || indicates the lines are not parallel. ________
Parallel Lines and Transversals Transversal A _____ is a _______ which intersects ____ or _____ lines in a plane. The intersected lines do not have to be parallel. Lines j, k, and m are intersected by line t. Therefore, line t is a ______ of lines j, k, and m.
Special Angle Relationships WHEN THE LINES ARE PARALLEL Exterior 1 2 3 4 Interior 5 6 7 8 Exterior If the lines are not parallel, these angle relationships DO NOT EXIST. ♥Alternate Interior Angles are _______ ♥Alternate Exterior Angles are ________ ♥Same Side Interior Angles are ________ ♥Same Side Exterior Angles are _________
Interior Angles The angles that lie in the area between the two parallel lines that are cut by a transversal, are called interior angles. L G Line L Exterior A 1200 P 1200 600 D B 600 F ________ Line N _________ Interior E Q Line M Exterior A pair of interior angles lie on the ______of the transversal.
Alternate Interior Angles Alternate angles are formed on opposite sides of the transversal and at different intersecting points. L A D G Line L P B Q E Line M Line N ____________ F ______of alternate angles are formed. Pairs of alternate angles are _______. | 677.169 | 1 |
The Elements of Plane Geometry ...
From inside the book
Results 1-5 of 18
Page 17 ... common vertex and common arms ) are said to be conjugate . The greater of the two is called the major conjugate , and the smaller the minor conjugate , angle . When the angle contained by two lines is spoken of with- out qualification ...
Page 22 ... common angle ABC c ; then the angle ABE is equal to the angle ABD , Ax . e . the part to the whole , Ax . a . which is impossible ; therefore BE is not in a straight line with BC . In the same way it can be shown that no other straight ...
Page 23 ... common angle AOD ; take away then the angle AOC is equal to the angle BOD . In the same way it may be proved that the angle BOC is Ax . e . equal to the angle AOD . Q.E.D. Ex . 3. The bisectors of two vertically opposite angles are in ...
Page 24 ... common arm . DEF . 25. The perimeter of a rectilineal figure is the sum of its sides . DEF . 26. A quadrilateral is a polygon of four sides , a pentagon one of five sides , a hexagon one of six sides , and so on . DEF . 27. A triangle ...
Page 38 ... common to both , therefore AE is equal to AF . Hyp . I. 12 . Hyp . I. 6 . Also , of AE and AF let AE be the one that is on the same side of the perpendicular as AG , then , in the triangle AGE , the angle AEG , being an exterior angle | 677.169 | 1 |
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Finding Angle Measures Parallel Lines Cut Transversal Worksheet
Finding Angle Measures Parallel Lines Cut Transversal Worksheet - For extra practice, students can write their own. Web angle pairs created by parallel lines cut by a transversal. Web when parallel lines are cut by a transversal, several different pairs are formed that are either congruent to each other (have the same angle measure) or the angle pairs might. 107° 81° 91° 69° 85° 91° 73° 79° 93° 81° 103° 95° 89°. Are you ready to keep your. Web looking for a fun and different way to practice angle measures with parallel lines cut by a transversal? Web this is a short quiz for finding angle measures with parallel lines and transversals. • use algebra to find unknown. Web this 8th grade angle relationships for parallel lines cut by a transversal worksheet is a great way for your students to analyze diagrams with a pair of parallel lines intersected. Only one pair of lines are parallel, so students must understand which angles pairs are.
Only one pair of lines are parallel, so students must understand which angles pairs are. The given parallel lines are cut by a transversal, therefore, the marked angles. • understand the parallel lines cut by a transversal theorem and it's converse. Web we discuss parallel lines, transversals and the various angles formed such as corresponding, alternate interior, alternate exterior, consecutive interior, and vertical. Web find the measure of the angle indicated in bold. Students are first asked to use their understanding of. Find the value of x in the given parallel lines 'a' and 'b', cut by a transversal 't'.
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Finding Angle Measures Parallel Lines Cut Transversal Worksheet
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50 Finding Angle Measures Worksheet
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Finding Angle Measures Parallel Lines Cut Transversal Worksheet
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Transversals of Parallel Lines Poly Ed
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Angles In Transversal Worksheet Answer Key inspirex
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Web this 8th grade angle relationships for parallel lines cut by a transversal worksheet is a great way for your students to analyze diagrams with a pair of parallel lines intersected. For each set of.
Parallel Lines Cut By A Transversal Worksheet Answer Key —
Web when parallel lines are cut by a transversal, several different pairs are formed that are either congruent to each other (have the same angle measure) or the angle pairs might. Students will then use.
Parallel Lines Cut by a Transversal Sample 1 ⋆
These worksheets feature questions on identifying the relationship between the given angles in parallel lines cut by a transversal. Only one pair of lines are parallel, so students must understand which angles pairs are. 107°.
Angle measuresparallellinescutbyatransversalppttmilsn6.01
Finding Angle Measures Parallel Lines Cut Transversal Worksheet - Web parallel lines and transversals worksheets can help students to learn about angles formed by parallel lines cut by a transversal. • find angle measures using the theorem. Are you ready to keep your. • understand the parallel lines cut by a transversal theorem and it's converse. Web looking for a fun and different way to practice angle measures with parallel lines cut by a transversal? Web this 8th grade angle relationships for parallel lines cut by a transversal worksheet is a great way for your students to analyze diagrams with a pair of parallel lines intersected.
For each set of angles name the angle pair, write the equation, solve the equation for x, and plug in x to find the. Web this is a short quiz for finding angle measures with parallel lines and transversals. Web parallel lines and transversals worksheets can help students to learn about angles formed by parallel lines cut by a transversal. Find the value of x in the given parallel lines 'a' and 'b', cut by a transversal 't'. Are you ready to keep your.
Web when parallel lines are cut by a transversal, several different pairs are formed that are either congruent to each other (have the same angle measure) or the angle pairs might. Web 6) if parallel lines cut by transversal, then coresponding angles congruent 7) substitution property reasons 1) given 2) if parallel lines cut by transversal, then coresponding. Web we discuss parallel lines, transversals and the various angles formed such as corresponding, alternate interior, alternate exterior, consecutive interior, and vertical. Web naming angle pairs formed by parallel lines cut by a transversal.
• Use Algebra To Find Unknown.
The given parallel lines are cut by a transversal, therefore, the marked angles. Web find the measure of the angle indicated in bold. Students are first asked to use their understanding of. Web looking for a fun and different way to practice angle measures with parallel lines cut by a transversal?
Are You Ready To Keep Your.
Web this is a short quiz for finding angle measures with parallel lines and transversals. Web we discuss parallel lines, transversals and the various angles formed such as corresponding, alternate interior, alternate exterior, consecutive interior, and vertical. • understand the parallel lines cut by a transversal theorem and it's converse. Web parallel lines and transversals worksheets can help students to learn about angles formed by parallel lines cut by a transversal.
These Worksheets Feature Questions On Identifying The Relationship Between The Given Angles In Parallel Lines Cut By A Transversal.
For each set of angles name the angle pair, write the equation, solve the equation for x, and plug in x to find the. Web when parallel lines are cut by a transversal, several different pairs are formed that are either congruent to each other (have the same angle measure) or the angle pairs might. Web angle pairs created by parallel lines cut by a transversal. Web this set of 24 task cards provides students with practice finding missing angle measures and applying the following angle relationships:corresponding anglesalternate interior. | 677.169 | 1 |
Geometry In High School: Real Life Applications Of Various Shapes
In Class 11 and 12 NCERTs, we explore straight lines and 3D shapes. We discuss lines, understanding their slopes, the angles between them, and different ways to write their equations.
This article explores how shapes and lines are part of our daily lives, helping us understand things better and making a big difference in many areas like technology, design, and medicine.
Geometry In Nature
Geometry is really important in our daily lives, especially when we look at nature. If we pay attention, we'll spot lots of shapes like circles, squares, and more in leaves, flowers, stems, and roots. Even our digestive system, like tubes inside tubes, shows how geometry matters. Trees have leaves of all sizes and shapes. And think about fruits and veggies; like an orange—it's round like a ball. When you peel it, see how the pieces fit together perfectly to make that round shape.
Geometry In High School: Real Life Applications Of Various Shapes
Studying a honeycomb up close reveals rows of hexagonal shapes. Similarly, observing a snowflake through a microscope allows us to marvel at its gorgeous geometric patterns.
Another fascinating instance of geometry in nature involves a pattern commonly referred to as the "Five-Around-One." This pattern is seen in flowers, known for their "Five-around-one" arrangement, which is also termed as "Closest Packing of Circles," "Pentagonal Packaging," and "Tessellating Pentagons."
Geometry In Technology
Geometry plays a massive role in our daily lives, especially in technology. Whether it's robotics, computers, or video games, geometry is fundamental to nearly all these innovations. Computer programmers rely on geometric concepts extensively in their work. For instance, the captivating virtual realms of video games come to life through intricate geometric calculations. Take raycasting, a technique used for simulating shooting in games. It involves using a 2-D map to create the illusion of a 3-D world, boosting the efficiency of processing by calculating vertical lines on the screen. This incorporation of geometry is what drives the complexity and realism we experience in these technological marvels.
Geometry In Homes
Geometry plays a vital role in our homes, influencing various aspects of everyday life. Just look around—windows, doors, beds, chairs, tables, TVs, mats, rugs, and cushions all come in different shapes. Additionally, bedsheets, quilts, covers, mats, and carpets showcase diverse geometric patterns. Even in cooking, geometry matters. Chefs rely on accurate proportions and ratios of ingredients to craft delicious dishes.
When organising a room, every space is utilised to enhance its appeal. Vases, paintings, and decorative pieces, with their assorted geometric shapes and patterns, contribute to making a house more inviting and presentable.
Geometry In Architecture
Geometry plays a vital role in creating buildings such as our new parliament and monuments. Before starting construction, mathematics, and geometry are used to plan how the building will be structured. The ideas of proportions and symmetries help outline the essential elements for all sorts of architectural designs. As early as the sixth century BC, Pythagoras' "Principles of Harmony" and geometry influenced architectural designs.
Integrating maths and geometry not only enhanced the beauty, balance, and spiritual significance of these grand structures but also helped reduce risks from strong winds. This combination of mathematical principles and geometric insights didn't just make buildings look good, it also made them safer.
Geometry In Design
Geometry is extensively used in various design fields. When it comes to creating animated characters in video games, geometry is crucial. In art, nearly every aspect of design is linked to geometric proportions, serving as a tool to convey stories. For instance, in miniature paintings and manuscript illumination, geometric principles are utilised to arrange the layout. Even in calligraphy, precise geometric proportions are carefully considered when forming each letter.
In design, geometry takes on a symbolic role, as seen in the intricate carvings adorning the walls, roofs, and doors of architectural wonders. These elements showcase how geometry plays a fundamental part in design across different creative disciplines.
Geometry In Computer Aided Design (CAD)
Geometry, a fundamental aspect of mathematics, involves exploring lines, curves, shapes, and angles. In architecture, computer software plays a crucial role by creating visual representations before any actual design work begins. CAD, a specialised software, generates the blueprint of the design. Furthermore, it assists in simulating architectural forms, giving a clearer picture of the final product.
The principles of geometry are extensively applied in various industries, facilitating the creation of detailed graphics and aiding in diverse industrial processes.
Geometry In Medicine
Methods such as x-rays, ultrasounds, MRIs, and nuclear imaging heavily rely on geometry to reconstruct the shapes of organs, bones, and tumours. Even in physiotherapy, geometry plays a significant role. Geometric properties and characteristics are pivotal in defining images within digital grids. These geometrical ideas don't just assist in visualising and manipulating images but also in segmenting, correcting, and representing objects. Moreover, they contribute to enhancing stability, accuracy, and efficiency in these processes. Techniques like bisecting angles and parallel methods are particularly crucial in radiology for accurate diagnosis and imaging successful production | 677.169 | 1 |
news
Understanding Step Angle in Stepper Motors
Release Date:2023-09-20
Author:LEESN
In the world of electromechanical devices, stepper motors play a pivotal role. One fundamental characteristic of stepper motors is the "Step Angle," which is crucial for their operation. But what exactly is the Step Angle, and how is it calculated? Let's delve into this essential concept.
Defining Step Angle:
The Step Angle, often denoted in degrees (°) or radians (rad), is the measure of the angle through which the rotor of a stepper motor rotates with each step. It determines the motor's ability to move and position accurately, making it a critical parameter in stepper motor specifications.
Calculating the Step Angle:
The calculation of the Step Angle depends on the type of stepper motor, and there are primarily two common types: single-phase and two-phase stepper motors.
1. Single-Phase Stepper Motors:
Single-phase stepper motors typically come with predetermined Step Angles, such as 50°, 72°, or 90°. These values are inherent to the motor's design and don't require additional calculation.
2. Two-Phase Stepper Motors:
For two-phase stepper motors, the Step Angle is usually calculated as 360° divided by the number of steps the motor can take in one full revolution. For instance, if you have a two-phase stepper motor with 200 steps per revolution, the Step Angle would be:
Step Angle = 360° / 200 = 1.8° per step
This means that the rotor moves 1.8 degrees with each step, and it takes 200 steps to complete one full revolution.
In summary, the Step Angle is a fundamental parameter that defines the precision and control of stepper motors. Understanding it and how it's calculated is essential for selecting the right motor for your application and ensuring precise motion control in various industries, from robotics to CNC machines. | 677.169 | 1 |
1 Answer
1
Find the plan that runs through the middle between them $C = \frac{P+P'}{2}$ and is perpendicular to the line connecting them.
Do this for all 3 of them and find the line of the planes' intersection. That will be the rotation axis.
If all mid-planes are coplanar then you can use the planes of the triangles themselves.
To get the angle you can take a $P$ and $P'$ again and project them onto the axis.
Take a point $A$ on the axis and the direction $v$ of the axis. The projected point is $P_p=A+\frac{v \cdot (P-A)}{v\cdot v} v$. And then the angle is $acos(\frac{dot(P-P_p, P'-P_p)}{|P-P_p|* |P'-P_p|})$
$\begingroup$2 is enough but the third lets you verify that it's indeed a rotation. As for angle you can project the point and it's rotated point onto the axis get the direction to the points and dot product.$\endgroup$
$\begingroup$I guess two can have degenerate cases if they're coplanar with the axis of rotation, so three can help, but three can also be bad if they're still coplanar with the axis. Like for a triangle on a door that rotates open, all the "midpoint-planes" are the same.$\endgroup$
$\begingroup$Great, also how do I actually do the project/find angle thing? How can I get two points to be in this new 2d coordinate system defined by the orthogonal plane to the rot-axis, so I can do a dot product?$\endgroup$ | 677.169 | 1 |
Show Square Offset
The Show Square Offset function displays the Offset distance of a point from a line.
If a centre point is clicked or entered, the square offset geometry is different.
See the diagram at the bottom of the page.
It will show the Arc Distances instead of the straight line distances.
The offset is from the 'FromPt' to the circular curve.
The straight line is through the centre point and the From Pt.
From Point
To select the offset or From point, click on a point on the screen, or enter the point number at the keyboard.
Type in the point number and press ENTER, TAB or Down Arrow to move to the next field.
Point 1
To select Point 1Point 2
To select Point 2Centre Point
Optional, click the centre point or enter it at the keyboard.
If a centre point is entered, the radii to points 1,2 will be checked and if within 20% of the radius,
the distances 1-Ip, IP-2 will be shown as arc distances.
Bearing 1 - 2
This field is the bearing of the line from Point 1 to Point 2 in (258°10'43") format.
Dist 1–IP
This is the distance from Point 1 to the Intersection Point (IP) in metres.
If the line 1-2 is a circular arc, this will be shown as the Arc Distance (Arc Dist 1-IP)
Dist IP-2
This is the distance from the Intersection Point (IP) to Point 2 in metres.
If the line 1-2 is a circular arc, this will be shown as the Arc Distance (Arc Dist IP-2)
Grid Offset
This is the offset distance of the From point to the line Pt1 – Pt2.
Ground Offset
If the job is on a Datum, this will show the Ground Offset distance
Note: All distances (1-IP, IP-2, Offset, Radius) are GRID distances except for this field
Left or Right
This indicates if the point is on the Left or the Right of the line Pt1 – Pt2.
Hold Pt1-Pt2
Tick this to hold the Point 1, Point2 and Centre Point values.
This makes it easy to click on successive 'From' points to show the square offset from a line.
Advanced
Tick this to display the 'Advanced version which retains a listing of successive Square Offsets.
Height Difference
It will compute the Height Difference between the From Point and the IP.
It shows the FromPtHeight - IPHeight.
All three points must have height values and the IP must be in between Point 1 and Point 2
Compute
Press this button to re-compute the square offset values and add them to this Listing
Clear
Press to clear the Listing and square offset values
Copy
Press to copy the listing to the clipboard.
You can then save to afile if necessary
Insert IP
This option will insert a new point athe the Intersection Point (IP) | 677.169 | 1 |
Exploring math.tan for Tangent Function
The tangent function, often denoted as tan, is one of the six fundamental trigonometric functions. In a right-angle triangle, the tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. Mathematically, if θ is an angle in a right triangle, and opposite and adjacent are the lengths of the sides opposite and adjacent to θ respectively, then:
tan(θ) = opposite / adjacent
In the unit circle, where the radius is 1, the tangent function can also be seen as the length of the line segment from the origin to a point on the tangent line to the circle at (1,0). As the angle θ increases, the point moves along the tangent line, and as such, the tangent function oscillates over its range.
The tangent function has a period of π radians (or 180 degrees), meaning that it repeats its values every π radians. It's an odd function, which implies that tan(-θ) = -tan(θ). The function is undefined at odd multiples of π/2 radians (or 90 degrees), where it approaches infinity or negative infinity. This behavior creates vertical asymptotes in the graph of the tangent function.
The tangent function has wide use in various fields including physics, engineering, and mathematics. It is essential for solving problems involving angles and distances such as in trigonometry, calculus, and even in complex number calculations. Understanding how to work with the tangent function is a fundamental skill for anyone delving into these areas.
Understanding the math.tan Function in Python
In Python, the math module provides a method called math.tan() which computes the tangent of a given angle. The angle must be in radians, not degrees. To convert degrees into radians, you can use the math.radians() function. Here is a simple example:
It's important to note that since the tangent function has vertical asymptotes, when using math.tan() with values that are close to these asymptotes (i.e., odd multiples of π/2), the result may be very large or negative, reflecting the behavior of the function as it approaches infinity. Here is an example:
This reflects the fact that the tangent function is approaching infinity as the angle approaches π/2 radians (90 degrees).
When working with the math.tan() function, it is crucial to handle these cases carefully, especially if your program needs to work with angles that might be close to the asymptotes.
Implementing the Tangent Function in Python
Now that we understand the basics of the tangent function and how to use the math.tan() function in Python, let's look at some practical implementations. One common use case is when calculating the slope of a line. In trigonometry, the slope of a line is equivalent to the tangent of its angle of inclination. Here's how you might implement this:
Another common application is when working with periodic functions in calculus or physics. Since the tangent function is periodic, it can be used to model oscillations or waves. For instance, you might use math.tan() to calculate the displacement of a wave at a given point in time:
It is important to remember that if time corresponds to an odd multiple of π/2, the displacement will be undefined since the tangent function will approach infinity.
When implementing math.tan(), handling exceptions is also essential. For example, you might want to catch cases where your program might inadvertently try to calculate the tangent of an angle where it is undefined. Below is an example of how you could handle such exceptions:
In this code snippet, we define a function safe_tangent() that wraps math.tan() in a try-except block. If calculating the tangent raises a ValueError, it prints a message and returns None. Otherwise, it returns the result of math.tan(). By using this function, we can avoid runtime errors due to undefined tangents.
To wrap it up, implementing the tangent function in Python using math.tan() is straightforward, but care must be taken around its vertical asymptotes. By understanding the behavior of the tangent function and handling exceptions properly, one can avoid common pitfalls and harness the power of this trigonometric function in various mathematical and scientific applications.
Exploring Applications of the Tangent Function
Aside from the examples mentioned above, the tangent function is also particularly useful in navigation and surveying. For instance, sailors use the tangent function to calculate their course when navigating on a spherical Earth. Similarly, surveyors might use it to determine the height of a building or a tree without needing to measure it directly. That is done by measuring the angle of elevation from a known distance and then applying the tangent function to find the height. Here's how such a calculation might look in Python:
The tangent function also plays an important role in computer graphics, particularly in 3D rendering. It is used to calculate angles of perspective and projection. For example, when determining the field of view for a camera in a 3D environment, you can use the tangent function to calculate the necessary angles based on the desired width and height of the view.
These are just a few examples of how the tangent function can be applied across different fields. Whether it is for solving geometric problems or modeling complex systems, understanding how to implement and utilize math.tan in Python allows for a wide range of practical applications.
Further Resources
The tangent function is an essential tool in various mathematical and scientific applications. However, it is important to remember that it has its limitations and peculiarities, such as the vertical asymptotes. By understanding these characteristics, programmers can ensure they use the math.tan() function effectively and avoid potential errors in their calculations.
If you're looking to deepen your understanding of the tangent function and its applications, there are many resources available. Online platforms like Khan Academy and Coursera offer courses in trigonometry and calculus that cover the tangent function in-depth. Additionally, textbooks on pre-calculus and calculus often have entire chapters dedicated to trigonometric functions, including tangent.
For those who prefer interactive learning, websites like Desmos offer graphing calculators that allow you to visualize the tangent function and its properties. Another great resource is Wolfram Alpha, which can compute tangent values and even show step-by-step solutions for more complex problems involving the tangent function.
In terms of Python-specific resources, the Python documentation for the math module is a good starting point. For a more hands-on approach, websites like Codecademy and HackerRank provide exercises and challenges that involve using the math.tan() function and other trigonometric functions in Python.
Remember that practice is key when learning any new mathematical concept. By exploring the resources mentioned above and writing your own Python programs that implement the tangent function, you'll solidify your understanding and be able to apply it confidently in your future projects. | 677.169 | 1 |
Elements of Geometry, Geometrical Analysis, and Plane Trigonometry: With an Appendix, Notes and Illustrations
From inside the book
Results 1-5 of 75
Page 2 ... distance ; and a surface represents extension . A line has only length ; a surface has both length and breadth ; and a solid combines all the three dimen- sions of length , breadth , and thickness . A The uniform description of a line ...
Page 3 ... distance , or linear extent , from progressive motion , we derive that of diver- gence , or angular magnitude , from revolving mo- tion . GEOMETRY is divided into Plane and Solid ; the former confining its views to the properties of ...
Page 11 ... distance G describe a circle , and from the centre B with the distance H describe an- other circle meeting the former in the point C : ACB is the tri- angle required . Because all the radii of the same circle are equal , AC is equal to ...
Page 12 ... distance DF ; and for the same reason , B must also be found in the circumference of a circle described from E , with K the distance EF : The vertex of the triangle ACB must , therefore , occur in a point which is common to both those ...
Page 15 ... distance DC de- scribe a circle ; and in the same line take another point E , and with distance EC describe a second circle inter- secting the former in F ; join CF , cross- ing the given line in G : CG is perpen- dicular to AB . For | 677.169 | 1 |
It is necessary for me to find midpoint of some arc in MATLAB. For simplicity let this arc is a part of circle. Also, we know the radius, center and starting point and ending point of the arc. I used $atan2(y,x)$ to get the angle for each line constructed by start and end points. Then find $t_{mid}=\frac{t_1+t_2}{2}$ and use $x_{mid}=r*cos(t_{mid})+xc,~y_{mid}=r*sin(t_{mid})+yc.$
But the problem is that $atan2$ gives the angle between $-\pi$ and $\pi$. Then for example for third quarter if $t_1=\pi$ and $t_2=-\pi/2$, $t_{mid}=\pi/4$ and it is not true. If I add $2\pi$ to the angles lower than zero, then I have the same problem for fourth quarter, for example $t_1=\frac{3\pi}{2},t_2=0$, the mid angle is $\frac{3\pi}{4}$!!. How can I solve this problem?
$\begingroup$The problem is not with $atan2$ but is in the identification and then evaluation of $t_{mid}$: Imagine both the endpoints and the center lie on the same line (let's say the x axis), how do you decide which is the arc to draw? Maybe your question can be reworded in find the midpoint of the smallest arc between two points, in this case, evaluate $\Delta t=t_1-t_2$, add or subtract $\pi$ until $|\Delta t|<\pi$ and then $t_{mid}=t_1-\Delta t/2$.$\endgroup$
1 Answer
1
Compute the coordinates of the midpoint $I$ of the endpoints (which is outside of the arc, of course): its polar angle is the same as the polar angle of the point you are looking for. The rest is easy. | 677.169 | 1 |
Class 8 Courses
ABC is a triangle in which altitudes BE and CF to sides AC and AB are equal (see the given figure). Show that<br/><br/>(i) $\triangle \mathrm{ABE} \cong \triangle \mathrm{ACF}$<br/><br/>(ii) $A B=A C$, i.e., $A B C$ is an isosceles triangle. | 677.169 | 1 |
to how the system is managed. Suppose we have upon the board the figure for the proposition: "If two triangles have two sides of the one equal respectively to two sides of the other and the included angles unequal etc." and the triangles with the smaller angle placed upon the other....
...joining the vertices of the two triangles is bisected by the line on which the bases stand. 89. If two triangles have two sides of the one equal respectively to two sides of the other, and the included angles supplementary, the triangles are equal in area. 90. If equilateral triangles...
...line BP. To Prove Z PBM = Z PBN. (', "i / PROP. XXIX. THEOREM 100. If two triangles have two sides of one equal respectively to two sides of the other,...first greater than the included angle of the second, the third side of the first is greater than the third side of the second. F JH G Draw A ABC. Construct ^ of the second, the third side of the first is > the third side of the...
...sides of the other. Now, maybe I can get along without those troublesome angles. Let me see. I had two sides of the one equal respectively to two sides of the other. I must show that DB is equal to DC. Stuck again. They do look equal, but there is nothing in my hypothesis...
...impossible : C B' Bare impossible: BBXXII 132. If two triangles have two sides of one equal respectively to two sides of the other, and the included angle of the first greater than the included angle of the second, the third side of the first is greater than the third side of the second. Draw &ABC and ELM, having... | 677.169 | 1 |
Protractor measure
While searching our database we found 1 possible solution for the: Protractor measure crossword clue. This crossword clue was last seen on March 19 2024 Wall Street Journal Crossword puzzle. The solution we have for Protractor measure has a total of 5 letters.
Answer
A
N
G
L
E
The word ANGLE is a 5 letter word that has 2 syllable's. The syllable division for ANGLE is: an-gle | 677.169 | 1 |
$\begingroup$I like this answer because it explains the math involved plus it gives two views of a solution in two different ways: ASCII art vs Graphic image AND Tilted vs Non-tilted square. I also appreciate the guide dots and the guidelines.$\endgroup$ | 677.169 | 1 |
Multiple Angle Formulas
Multiple Angle Formulas is a very important Mathematical concept. It is part of Trigonometry. The area of mathematics known as trigonometry is concerned with the mathematical properties of specific angles and how to employ those properties in computations. There are six widely used angles-related trigonometric functions. Their respective names and acronyms are sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc). These six trigonometric functions are depicted in the figure in relation to a right triangle. The sine of the angle, or sin A, for example, is the ratio between the side opposite to an angle and the side opposite to the right angle (the hypotenuse); other trigonometric functions are defined similarly. Before computers rendered trigonometry tables obsolete, these functions—which are characteristics of the angle, regardless of the size of the triangle—were listed for numerous angles. In geometric figures, unknown angles and distances are derived from known or measured angles using trigonometric functions.
The need to calculate angles and distances in disciplines like astronomy, map making, surveying, and artillery range finding led to the development of trigonometry. Plane trigonometry deals with issues involving angles and lengths in a single plane. Spherical trigonometry takes into account applications to similar issues in more than one plane of three-dimensional space.
Applications: Trigonometrical applications can be found in many areas of daily life. Astronomy is one of the well-known fields where trigonometry is used to calculate the distances between the Earth and other planets and stars. It is used to create maps for navigation and geography. Finding an island's location in relation to its longitude and latitude uses applications of trigonometry as well. Even today, some of the most cutting-edge techniques employed in engineering and the physical sciences are founded on trigonometric ideas.
Trigonometry is used in a variety of fields, including surveying, astronomy, and building. The two most prevalent subjects are astronomy and physics, in which it is used to measure the separation between planets and stars, determine their velocity, and analyse waves. Among the applications are: Oceanography, seismology, meteorology, physical sciences, astronomy, acoustics, navigation, electronics, and a host of other disciplines. It is also useful to calculate the length of long rivers, gauge a mountain's height, etc.
Astronomers have utilised spherical trigonometry to determine the positions of the sun, moon, and stars. Trigonometry is also employed in the following areas:
Trigonometry is necessary for criminology or crime scenes to ascertain the approximate cause of a collision in a car accident or at what angle a gunshot was fired. Trigonometry is used by marine scientists to determine the depth of sunlight that influences algae's capacity for photosynthesis. Waves of sound and light can be described using trigonometric functions.
Oceanographic wave and tidal height calculations are made using trigonometry.
Algebra and trigonometry are the foundations of calculus.
Multiple Angle Formulas
Trigonometric functions frequently contain Multiple Angle Formulas. Although it is impossible to directly obtain the values of Multiple Angle Formulas, their values can be calculated by expanding each trigonometric function. Multiple Angle Formulas is another name for the multiple-angle trigonometric function. The many angle formulas employ the double and triple angles formulas. The common functions used for the Multiple Angle Formulas include sine, tangent, and cosine.
Trigonometric functions frequently contain Multiple Angle Formulas. Although it is impossible to directly obtain the values of Multiple Angle Formulas, their values can be calculated by expanding each trigonometric function.
Multiple Angle Formulae
When the term "Multiple Angle Formulas" is used to describe trigonometric functions with multiple angles. Multiple Angle Formulas include double, triple, and half-angle formulas.
List of Multiple Angle Formulae
Half Angle Formula
Double Angle Formula
Triple Angle Formula
It is common for trigonometric functions to contain Multiple Angle Formulas. Despite the fact that it is impossible to directly obtain their values, their values can be calculated by expanding each trigonometric function. Mathematical concepts like this are very important. Essentially, it's a part of trigonometry. Known as trigonometry, this field of Mathematics focuses on the properties of angles and how to use them. One can study the Multiple Angle Formulas on the Extramarks website or mobile application. One is required to register on the website of Extramarks in order to be able to access the learning resources provided on the concerned website.
Generalized Multiple Angle Formulae
Students can learn more about Generalised Multiple Angle Formulas from the various resources provided by Extramarks. Extramarks provides resources in accordance with the latest NCERT syllabus. All the resources provided are prepared after consideration of past years' question papers. Further, the resources are written in an easy-to-understand language. Students can rely on the resources provided by Extramarks as they are written by expert subject teachers and are proofread regularly.
Sample Problems on Multiple Angle Formulas
To understand any topic better, students should solve practice problems on it. To understand the topic of Multiple Angle Formulas well, students can solve questions based on it. The practice questions will help students to apply their knowledge. If they face any difficulty in solving questions of Multiple Angle Formulas, they can refer to the resources offered by Extramarks. Extramarks provided resources are of high quality. The resources such as revision notes help students strengthen their basics.
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FAQs (Frequently Asked Questions)
1. What are Multiple Angle Formulas?
Trigonometric functions frequently contain the Multiple Angle Formulas. Although it is impossible to directly obtain the values of Multiple Angle Formulas, their values can be calculated by expanding each trigonometric function.
2. From where can students download the resources offered by Extramarks?
Students can download the various resources provided by Extramarks from their website and mobile application. | 677.169 | 1 |
Geometry
Top 173 CAT Geometry Questions With Video Solutions [PDF]
Geometry is an important and challenging topic in the CAT exam with a lot of weightage. CAT Geometry questions are asked from topics including Triangles, Circles, Quadrilaterals, Polygons and so on. To perform Geometry, one must thoroughly understand concepts, and formulas and gain problem-solving skills. Previous years' question papers are invaluable resources for CAT preparation, especially for mastering the CAT Quantitative Aptitude section. By solving the questions from CAT's previous papers, candidates can understand the types of questions asked, the level of difficulty, and the exam pattern. Below, you can find all those Geometry questions separated year-wise, along with the video solution for each question. Keep practising free CAT mocks where you'll get a fair idea of how questions are asked, and type of questions asked of CAT Geometry Questions. Also, you can download all the below questions in a PDF format consisting of video solutions for every problem explained by the CAT experts. Click the link below to download the CAT Geometry Questions with video Solutions PDF.
CAT Geometry Topic-wise Weightage Over Past 6 Years
CAT Geometry Formulas PDF
CAT Geometry is one of the most important topics in the quantitative
aptitude section, and it is vital to have a clear understanding of the
formulas related to them. As mentioned earlier, there will be high weightage for this concept if you can compare the past few CAT question papers. To help the aspirants to ace this topic, we have
made a PDF containing a comprehensive list of formulas, tips, and
tricks that you can use to solve Geometry questions with ease
and speed. Click on the below link to download the CAT Geometry formulas PDF.
1. Tangents on a circle
Tangents:
Direct common tangent: $$PQ^2=RS^2=D^2-\left(r_1-r_2\right)^2$$, where D is the distance between the centres:
Transverse common tangent: $$PQ^2=RS^2=D^2-\left(r_1+r_2\right)^2$$, where D is the distance between the centres:
2. Area, inradius, circumradius of triangles
If x is the side of an equilateral triangle then the
Altitude (h) =$$\frac{\sqrt{\ 3}}{2}x$$
Area =$$\frac{\sqrt{\ 3}}{4}x^2$$
Inradius = $$\frac{1}{3}\times\ h$$
Circumradius = $$\frac{2}{3}\times\ h$$
▪ Area of an isosceles triangle =$$\frac{a}{4}\sqrt{\ 4c^2-a^2}$$ (where a, b and c are the length of the sides of BC, AC and AB respectivelyand b = c)
CAT 2023 Geometry questions
Question 1
A triangle is drawn with its vertices on the circle C such that one of its sides is a diameter of C and the other two sides have their lengths in the ratio a : b. If the radius of the circle is r, then the area of the triangle is
Question 2
Let C be the circle $$x^{2} + y^{2} + 4x - 6y - 3 = 0$$ and L be the locus of the point of intersection of a pair of tangents to C with the angle between the two tangents equal to $$60^{\circ}$$. Then, the point at which L touches the line $$x$$ = 6 is
Question 3
Let $$\triangle ABC$$ be an isosceles triangle such that AB and AC are of equal length. AD is the altitude from A on BC and BE is the altitude from B on AC. If AD and BE intersect at O such that $$\angle AOB = 105^\circ$$, then $$\frac{AD}{BE}$$ equals
Question 6
In a rectangle ABCD, AB = 9 cm and BC = 6 cm. P and Q are two points on BC such that the areas of the figures ABP, APQ, and AQCD are in geometric progression. If the area of the figure AQCD is four times the area of triangle ABP, then BP : PQ : QC is
Question 7
Question 8
In a right-angled triangle ∆ABC, the altitude AB is 5 cm, and the base BC is 12 cm. P and Q are two points on BC such that the areas of $$\triangle ABP, \triangle ABQ$$ and $$\triangle ABC$$ are in arithmetic progression. If the area of ∆ABC is 1.5 times the area of $$\triangle ABP$$, the length of PQ, in cm, is
Question 6
Question 7
In a triangle ABC, AB = AC = 8 cm. A circle drawn with BC as diameter passes through A. Another circle drawn with center at A passes through Band C. Then the area, in sq. cm, of the overlapping region between the two circles is
Question 3
Frequently Asked Questions
How can video solutions help me in preparing for CAT geometry questions?
Video solutions can be a helpful resource for candidates preparing for CAT geometry questions. They can provide a step-by-step explanation of how to solve the problem, helping candidates better understand the concept and formula.
What are the main topics in geometry for CAT?
Triangles, Circles, Quadrilaterals, Polygons, 3-D Geometry and Coordinate Geometry are the most important topics that candidates should have well understanding. They should practice a wide range of questions related to each topic to excel in the Geometry topic in the CAT quant section. | 677.169 | 1 |
NCERT solutions for class 7 maths chapter 10 Practical Geometry
Simplify Practical Geometry with Free PDF Downloads of NCERT Solutions for Class 7 Math Chapter 10, "Practical Geometry," designed to make the understanding of geometry problems accessible for students. At [Your School Name], we have meticulously crafted these solutions to facilitate students in cracking problems with ease. Our step-by-step explanations ensure students can grasp the concepts thoroughly. Our NCERT Solutions aim to empower students to excel in geometry and develop confidence in solving practical geometry problems.
Exercise 10.1
Draw a line, say AB, take a point C outside it. Through C, draw a line parallel to AB using a ruler and compasses only.
Answer :
Steps for construction,
1. Draw a line AB.
2. Take any point Q on AB and a point P outside AB and join PQ.
3. With Q as the center and any radius, draw an arc to cut AB at E and PQ at F.
4. With P as the center and the same radius, draw an arc IJ to cut QP at G.
5. Place the pointed tip of the compass at E and adjust the opening so that the pencil tip is at F.
6. With the same opening as in step 5 and with G as the center, draw an arc cutting the arc IJ at H.
7. Now, join PH to draw a line CD.
Question 2 :
Draw a line L. Draw a perpendicular to L at any point on L. On this perpendicular choose a point X, 4 cm away from l. Through X, draw a line m parallel to L.
Answer :
Steps for construction,
1. Draw a line L.
2. Take any point P on line L.
3. At point P, draw a perpendicular line N.
4. Place the pointed tip of the compass at P and adjust the compass up to the length of 4 cm. Draw an arc to cut this perpendicular at point X.
5. At point X, again draw a perpendicular line M.
Question 3 :
Let
Answer :
Steps for construction,
1. Draw a line L.
2. Take any point Q on L and a point P outside L and join PQ.
3. Make sure that angles at point P and point Q are equal, i.e. ∠Q = ∠P
4. At point P, extend the line to get line M which is parallel to L.
5. Then take any point R on line M.
6. At point R, draw an angle such that ∠P = ∠R
7. At point R, extend the line which intersects line L at S and draw a line RS.
Exercise 10.2
Question 1 :
Construct ΔXYZ in which XY = 4.5 cm, YZ = 5 cm and ZX = 6 cm.
Answer :
Steps of construction
1. Draw a line segment YZ = 5 cm.
2. With Z as a centre and radius 6 cm, draw an arc.
3. With Y as a centre and radius 4.5 cm, draw another arc, cutting the previous arc at X.
4. Join XY and XZ.
Then, ΔXYZ is the required triangle.
Question 2 :
Construct an equilateral triangle of side 5.5 cm.
Answer :
Steps of construction
1. Draw a line segment AB = 5.5 cm.
2. With A as a centre and radius 5.5 cm, draw an arc.
3. With B as a centre and radius 5.5 cm, draw another arc, cutting the previous arc at C. | 677.169 | 1 |
Which Shows the Pre-image of Triangle X'y'z' Before the Figure Was Rotated 90° About the Origin?
Which Shows the Pre-image of Triangle X'y'z' Before the Figure Was Rotated 90° About the Origin?: In the field of geometries, our understanding the concept of transformations play a crucial role in understanding the manipulation of figures and shapes. One of these transformations, called rotation, is the process of rotating a figure around fixed point.
This article explores the concept of pre-images with a particular focus on the pre-image of the triangle that is x'y'z', before it undergoes 90deg of rotation around the origin. Through a thorough explanation of the basic principles and techniques involved readers will be able to gain an in-depth understanding of this concept.
Understanding the Basic Concepts:
Before diving into the nuances of pre-images and rotation it is essential to develop an understanding of the fundamental geometric concepts.
Triangle The term "triangle" refers to an equilateral polygon that has the three edges as well as three vertices. It is among the basic shapes in Euclidean geometry. It is distinguished through its angles and 3 edges.
Rotation: A transformation that revolves the figure around a fixed location, also known as"the center of the rotation. The figure is congruent (i.e. it retains its dimensions and shape) during the entire rotation.
Pre-image: The image of a person is the original orientation or position prior to undergoing the transformation. It is the beginning point of any transformation.
Understanding the Effects of Rotation:
The rotation can occur in different ways (clockwise and counterclockwise) and at different locations. However, in this case we will focus on a 90deg rotation around the origin.
When a figure goes through an 90deg turn around the origin in a counterclockwise direction, every point in the figure rotates 90deg about the origin. In clockwise rotation the points rotate 90deg opposite to each other.
Which Shows the Pre-image of Triangle X'y'z' Before the Figure Was Rotated 90° About the Origin?
Important Points to Take Note of:
To fully comprehend the pre-image of the triangle x'y'z' prior to rotation, a few key aspects must be considered:
The Orientation of the Triangle The initial position of the triangle x'y'z' occupies particular position on the plane of coordinates. The vertex (x', y") has distinct coordinates which determine the location of its vertex in relation to its origin.
Angle of Rotation A 90deg angle means that every point in the triangle will rotate 90 degrees around the origin. The rotation could be clockwise or counterclockwise, based on the direction that is specified.
Coordinates of the Vertices These coordinates on the vertices (x' and z')') of triangle x'y'z are crucial to know regarding its pre-image. The coordinates undergo a transformation depending upon the orientation and the angle of the rotation.
Examining the Pre-Image
To find the pre-image of triangle x'y'z' prior the rotation, we must take into account its initial position and the impact of a 90deg incline rotation the origin.
The original position: Triangle x'y'z' is initially located on the plane of coordinates with its vertices positioned at particular coordinates (x' and y')'). These coordinates determine its initial image before any transformation takes place.
Effects of 90deg Rotation 90deg of rotation about the origin is a process of shifting each vertex of the triangle by 90 degrees in a predetermined direction. This rotates the position of the triangle, while preserving its size and shape.
Coordinate Transformation Utilizing the laws of rotation, we can find the new coordinates for each vertex after 90deg rotation. These transformed coordinates show the appearance of the triangle following the rotation.
Illustration and Visualization: Graphical representations like diagrams or grids of coordinates, can assist in visualizing the image of the triangle x'y'z' prior the rotating. By drawing the original vertices, and showing the rotation, we are able to better comprehend the process of transformation.
Consider an example that illustrates the calculations of the pre-image the triangle the x'y'z' prior to rotation.
Vertice coordinates given to vertices These coordinates include the coordinates of vertices are: x' (2, 3) and 3y' (4, 5) and z' (6, 1)
Conclusion:
Knowing the notion of pre-image prior to the rotation is vital to comprehend geometric transformations efficiently. By examining the origin of a shape and the effect of rotation, we can figure out the pre-image of the figure with a high degree of accuracy. In the case of a triangle x'y'z', understanding its pre-image prior to 90deg of rotation around its origin requires careful examination of coordinates and fundamentals of rotation. By systematically analyzing and calculating we can discover how the initial image is formed by the triangular shape and increase the understanding about geometric transforms.
FAQs on which shows the Image of the Triangle X'y'z" Prior to the Figure was rotated 90deg about the Source?
What is a pre-image geometry?
Pre-images refer to the initial position and orientation of the figure prior to it changing through transformations such as rotation, translation or reflection.
How is the pre-image for an arc determined prior to its the rotation?
Pre-images of triangular shape prior to the rotation is determined by taking into account its initial position on the plane of coordinates and the effect of the rotation. The coordinates of the vertices offer essential information for determining the pre-image.
What exactly does 90deg incline in relation to the origin what does it mean?
90deg of rotation around the origin is the process of turning the figure 90 degrees around its origin in either counterclockwise or clockwise. This rotates the location of every point in the figure, while preserving its size and shape.
What happens when a 90deg turn alter the coordinates of a triangle's vertex?
When a 90deg rotate is made around the origin and the vertex coordinates of a triangle undergo transformation. Each vertice moves 90deg in the direction specified in relation to the origin which results in new coordinates for vertex that has been rotated.
What are the most important factors to consider when determining the pre-image of the triangle x'y'z' prior its rotation?
The most important thing to consider is identifying the initial coordinates of the triangle's vertex points (x', the y's, the z's) knowing the direction and the angle of rotation and utilizing the rules of transformation to determine the prior-image coordinates.
The pre-image for the triangle x'y'z' appear on an axis plane?
Yes the pre-image of triangle x'y'z' could be drawn on a coordinate plane making a plot of its original vertices, and illustrating the impact of the rotation. Graphical representations help in understanding the process of transformation.
What can I do to verify the accuracy of the pre-image coordinates I calculated?
You can confirm the accuracy of the pre-image coordinates you calculated by applying the transform rules in a consistent manner and then checking if the vertices rotated are aligned with the expected positions following the rotation.
Are there alternative ways to determine the pre-image the triangle x'y'z' prior the rotation?
The method described here involves using the coordinates of the pre-image directly other methods, such as using geometric software can also be employed to determine the pre-image prior to rotation.
What implications can knowing the pre-image of a triangle x'y'z' prior to rotation in real-world applications?
The understanding of the preimage the triangle x'y'z' prior to rotation is crucial in many areas like engineering, computer graphics and architecture in which geometric transformations play a significant part in the design and analysis. | 677.169 | 1 |
Home/Relationship/What is the Relationship between the Slopes of Perpendicular Lines
What is the Relationship between the Slopes of Perpendicular Lines
Updated onJanuary 16, 2024
Perpendicular lines have slopes that are negative reciprocals of each other. Therefore, if two lines are perpendicular, the slope of one line is the negative reciprocal of the slope of the other line.
The relationship between slopes of perpendicular lines is a fundamental concept in geometry and trigonometry. This relationship is based on the fact that perpendicular lines have opposite reciprocal slopes. In other words, if a line has a slope of m, then a line perpendicular to it will have a slope of -1/m.
This relationship is important in many areas of mathematics, including calculus, where it is used to calculate the slope of the tangent line to a curve at a given point. The relationship between perpendicular slopes is also crucial in physics, engineering, and other sciences that involve calculations related to angles and distances. Overall, understanding the relationship between slopes of perpendicular lines is essential for solving a wide range of mathematical problems in these fields.
Understanding Slopes And Perpendicularity
Definition Of Slopes
Understanding the relationship between the slopes of perpendicular lines is fundamental in geometry. A slope refers to the steepness of a line. It is calculated using the change in y divided by the change in x, or as rise over run.
The slope of a line can be positive, negative, zero or undefined.
A positive slope moves up and to the right, while a negative slope moves down and to the right.
A slope of zero indicates a horizontal line.
An undefined slope indicates a vertical line.
Definition Of Perpendicularity
Two lines are perpendicular if they intersect at a right angle. In other words, their slopes are negative reciprocals. That is, when you multiply one slope by the other, the answer will be negative one. For example, a line with a slope of 2/3 will be perpendicular to a line with a slope of -3/2.
Here are a few examples to further clarify slopes and perpendicularity:
Two vertical lines are always perpendicular.
Two horizontal lines can never be perpendicular to each other.
A vertical line is perpendicular to every horizontal line, and a horizontal line is perpendicular to every vertical line.
Examples To Further Clarify Slopes And Perpendicularity
Here are some examples of how to find the slope of a line and determine if two lines are perpendicular:
Example 1: Find the slope of the line passing through the points (2, 5) and (-3, 4).
Slope = (Y2 – Y1) / (X2 – X1)
= (4 – 5) / (-3 – 2)
= -1/5
The slope of the first line is 2, and the slope of the second line is -1/2. To find out if they are perpendicular, we need to multiply these slopes together and see if the result is -1.
2 * (-1/2) = -1, so the lines are perpendicular to each other.
Example 3: Find the slope of the line that is perpendicular to the line passing through the points (-1, 2) and (4, 8).
First, we need to find the slope of the line passing through those two points.
Slope = (8 – 2) / (4 – (-1))
= 6/5
The perpendicular slope will be the negative reciprocal of 6/5, which is -5/6.
The Relationship Between Perpendicular Lines
The Concept Of Perpendicularity
Perpendicular lines are two lines that intersect each other at a right angle. In other words, they form a 90-degree angle at their intersection point. The concept of perpendicularity is prevalent in mathematics and real-life applications, such as construction and design.
Properties Of Perpendicularity
Here are some properties of perpendicularity that you need to know:
Perpendicular lines always meet at a 90-degree angle.
The slope of one line is the negative reciprocal of the other. In other words, if m1 is the slope of the first line, then the slope of the second line (m2) is -1/m1.
Perpendicular lines have opposite signs/ direction.
Illustrative Examples Of Perpendicularity
Let's look at some examples of perpendicular lines:
The x-axis and y-axis in a 2-dimensional coordinate plane are perpendicular to each other.
The corners of a rectangle or a square form perpendicular lines.
Perpendicular bisectors of a triangle meet at a common point, called the circumcentre.
In architecture and construction, perpendicular lines ensure that corners are square and walls meet at right angles.
How To Calculate The Slopes Of Perpendicular Lines
To calculate the slope of a perpendicular line, follow these steps:
Find the slope of the first line (let's call it m1).
Take the negative reciprocal of m1. That is, flip the fraction and change the sign.
The result is the slope of the perpendicular line (m2).
For instance, let's suppose the slope of line a is -2/3. To find the slope of line b, which is perpendicular to line a, we take the negative reciprocal of -2/3 and obtain 3/2. Therefore, the slope of line b is 3/2.
Perpendicular lines play a significant role in various fields such as engineering, architecture, graphics, and physics. By knowing the properties and how to calculate the slope of perpendicular lines, we can solve many real-world problems.
Deriving The Slopes Of Perpendicular Lines
The Formula For Slopes Of Lines
Before diving into the topic of the relationship between slopes of perpendicular lines, let us review the formula for finding the slope of a line.
Slope, represented by the letter m, is defined as the ratio of the change in the y-coordinates to the corresponding change in x-coordinates.
The formula for slope can be expressed as: M = (y2 – y1) / (x2 – x1), where (x1,y1) and (x2,y2) are two points on the line.
It is important to keep in mind that if the points lie on a horizontal line, the slope is zero, and if the points lie on a vertical line, the slope is undefined.
Moving on to the main topic, let us understand the relationship between the slopes of perpendicular lines.
Perpendicular lines are lines that intersect at a 90-degree angle, forming a right angle.
When two lines are perpendicular, the product of their slopes is -1. In other words, if m1 is the slope of one line and m2 is the slope of the other line, then m1 m2 = -1. This property allows us to easily find the slope of one line if we know the slope of the other line.
To derive the slope of a line that is perpendicular to another line, we simply take the negative reciprocal of the slope of the first line. That is, if the slope of the first line is m1, then the slope of the perpendicular line is -1/m1.
Examples To Strengthen Conceptual Understanding
To strengthen our conceptual understanding, let us go through a few examples.
Example 1: Find the slope of a line perpendicular to the line y = 2x + 1.
Solution: The given line has a slope of m1 = 2. Therefore, the slope of the perpendicular line would be m2 = -1/2 (negative reciprocal of 2).
Example 2: Find the slope of a line that is perpendicular to the line passing through the points (2,3) and (5,-1).
Solution: The slope of the given line can be calculated using the slope formula:
M1 = (y2 – y1) / (x2 – x1) = (-1 – 3) / (5 – 2) = -4/3
Therefore, the slope of the perpendicular line would be m2 = -1 / (-4/3) = 3/4.
By now, you should have a good understanding of the relationship between slopes of perpendicular lines and how to derive the slope of a perpendicular line. Keep these concepts in mind as you move forward with your studies in geometry and algebra.
Practical Applications Of Perpendicular Lines
Perpendicular lines have a unique relationship that is useful in several fields. This section will discuss the practical applications of perpendicular lines in architecture and design, engineering, mathematics and physics, and their industry connections and implications.
Architecture And Design
Architects and designers use perpendicular lines to create structures with stability and balance. Some of the practical applications of perpendicular lines include:
Building walls that intersect each other at right angles, providing stability to the structure.
Creating stable and durable foundations for buildings and other structures.
Designing doorways and windows that are square or have right angles.
Engineering
Perpendicular lines play an essential role in engineering, where stability and balance are critical. Engineers use perpendicular lines in the following ways:
Designing bridges and other structures with stable foundations that can withstand heavy loads.
Creating roads that intersect each other at right angles, making intersections safer.
Designing machines that depend on precise angles for their efficient operation.
Mathematics And Physics
Perpendicular lines form the basis of geometry in mathematics since they represent right angles. They are also essential in physics, where the angle between two surfaces determines their relationship. Examples of their practical applications are:
Calculating the distance between two points in mathematics, using the pythagorean theorem.
Determining the angle of incidence and reflection in optics, where light passes through perpendicular lines.
Calculating work done and force in physics while using machines and electronic devices with perpendicular components.
Industry Connections And Implications
Perpendicular lines have significant implications in many industries, including construction, transportation, and manufacturing, among others. A few examples of their practical applications include:
The use of perpendicular lines in the layout of roads, railways, and other transportation infrastructure.
The importance of perpendicular lines in manufacturing and assembly lines, where precise angles are needed for production.
The application of perpendicular lines in the construction of buildings and other structures, providing stability and balance to the final project.
Overall, perpendicular lines are an essential aspect of geometry, physics, and many real-world activities. They provide the foundation for many practical applications across various fields, such as architecture, engineering, and manufacturing, among others.
Is Understanding Customer Relationships Similar to Understanding the Relationship Between Perpendicular Lines?
Understanding customer relationships requires the same precision as understanding the relationship between perpendicular lines. A clear customer relationship manager definition is essential for establishing and maintaining strong connections with clients, just as understanding the mathematical concept is crucial for solving problems in geometry. Both require careful attention to detail.
Frequently Asked Questions On What Is The Relationship Between The Slopes Of Perpendicular Lines
How Are The Slopes Of Perpendicular Lines Related?
The slopes of perpendicular lines are negative reciprocals of each other.
What Is The Significance Of The Slope Of A Line?
The slope of a line describes the steepness or incline of the line.
How Do You Find The Slope Of A Line?
To find the slope of a line, divide the change in y-coordinate by the change in x-coordinate.
Can Perpendicular Lines Have The Same Slope?
No, perpendicular lines cannot have the same slope.
How Can The Relationship Between Perpendicular Lines Be Applied In Real Life?
Perpendicular lines are used in construction, architecture, and engineering to create right angles and stable structures.
Conclusion
As we wrap up our exploration of the relationship between the slopes of perpendicular lines, we now have a clearer understanding of the important role slope plays in the study of geometry. A line's slope not only helps us determine its slope-intercept form, it also helps us identify whether other given lines are parallel or perpendicular.
By utilizing the concept of negative reciprocal slopes, we can easily find the slope of a perpendicular line. Understanding these fundamental concepts can prove to be crucial in tackling more complex geometric problems. As we continue to expand our knowledge in mathematics and geometry, it is important to never lose sight of the foundational concepts that make up the backbone of these subjects.
Armed with a stronger understanding of perpendicular line slopes, we can continue to build upon these concepts to reach greater mathematical heights. | 677.169 | 1 |
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Circle Calculator
Circle definition
A circle is a round plane figure with no corners or edges. It is a two-dimensional closed curve, made of points that are at equal distance from a given point – the center.
The name circle derives from the Greek κίρκος/κύκλος (kirkos/kuklos), meaning hoop or ring.
Parts of a circle
The four main parts of a circle are radius (r), diameter (d), area (A) and circumference (c).
Radius (r)
The radius of a circle is the distance from the center of the circle to any point on it's circumference.
It can be found with the following formulas:
If you know the diameter of the circle:
r = d / 2
If you know the circumference:
r = c / (2 * π)
If you know the area:
r = √(A / π)
Diameter (d)
The diameter of a circle is the length of a line that passes through the center and joins two points on the circle.
You can determine it with these formulas:
If you know the radius of the circle:
d = 2 * r
If you know the circumference:
d = c / π
If you know the area:
d = 2 * √(A / π)
Area (A)
The area of a circle is the region enclosed inside the circle.
You can calculate it as follows:
If you know the radius or diameter of the circle:
A = π * r² = π * (d / 2)²
If you know the circumference:
A = c² / (4 * π)
Circumference (c)
The circumference of a circle is the distance around the boundary of that circle.
You can calculate it with the following formulas:
If you know the radius or diameter of the circle:
c = 2 * π * r = π * d
If you know the area:
c = 2 * √(π * A)
Unit circle
The unit circle is a circle with a radius of 1. Usually, its center is positioned at the origin (0, 0) in the Cartesian coordinate system. The length from the center to any point on the circle is of length 1.
By choosing any point (x, y) on the unit circle, we have x and y as the lengths of the right triangle, with the radius of our circle being the hypotenuse of length 1.
Using the Pythagorean theorem, we can find out that:
x² + y² = 1
This equation works for every quadrant, since:
x² = (-x)²
We can also define the right triangle in terms of sine and cosine:
cos(α) = x / 1 = x
sin(α) = y / 1 = y
which results in:
cos²(α) + sin²(α) = 1
This equation is known as the Pythagorean trigonometric identity.
Pi (π)
The number π is a mathematical constant, defined as the ratio of a circle's circumference to its diameter. Regardless of the circle's size, the ratio c / d is constant.
π = circumference / diameter
The number π is an irrational number and its digits never repeat. Pi has been calculated to one hundred trillion digits beyond its decimal point (2022), however this constant is an infinite decimal. | 677.169 | 1 |
sin a+b
Table of Contents
Mathematics is a fascinating subject that encompasses various concepts and formulas. One such concept is "sin a+b," which plays a crucial role in trigonometry. In this article, we will delve into the depths of this concept, exploring its definition, properties, and applications. By the end, you will have a clear understanding of "sin a+b" and its significance in the world of mathematics.
Understanding "sin a+b"
Before we dive into the intricacies of "sin a+b," let's first establish its definition. In trigonometry, "sin a+b" refers to the sum of two angles, a and b, expressed in terms of the sine function. Mathematically, it can be represented as:
sin(a + b) = sin a * cos b + cos a * sin b
This formula allows us to calculate the sine of the sum of two angles by utilizing the sines and cosines of the individual angles. By understanding this formula, we can explore the properties and applications of "sin a+b" in greater detail.
Properties of "sin a+b"
The concept of "sin a+b" possesses several properties that make it a valuable tool in trigonometry. Let's examine some of these properties:
Property 1: Commutative Property
The commutative property states that the order of addition does not affect the result. Therefore, "sin a+b" is equal to "sin b+a." This property allows us to rearrange the angles while calculating the sum of their sines.
Property 2: Periodicity
The sine function is periodic with a period of 2π. Consequently, "sin a+b" exhibits the same periodicity. This property enables us to determine the values of "sin a+b" for any given angles within a specific range.
Property 3: Symmetry
The sine function is an odd function, meaning that "sin(-x)" is equal to "-sin x." As a result, "sin a+b" can be rewritten as "sin a+(-b)" or "sin a-b." This symmetry property allows us to simplify calculations and manipulate the formula accordingly.
Property 4: Addition Formula
The addition formula for "sin a+b" is derived from the trigonometric identities. It states that "sin a+b" can be expressed as the sum of the products of sines and cosines of the individual angles. This formula is crucial in solving complex trigonometric equations and simplifying expressions.
Applications of "sin a+b"
The concept of "sin a+b" finds numerous applications in various fields, including physics, engineering, and computer science. Let's explore some of its practical applications:
Application 1: Wave Analysis
In physics, waves play a fundamental role in understanding various phenomena. The concept of "sin a+b" allows us to analyze and manipulate waveforms, enabling us to study the interference and superposition of waves. By applying the addition formula, we can determine the resulting waveforms when two or more waves interact.
In the field of signal processing, the concept of "sin a+b" is utilized to analyze and manipulate signals. By decomposing signals into their frequency components using Fourier analysis, engineers can apply the addition formula to combine or modify signals, leading to advancements in telecommunications, audio processing, and image processing.
Application 4: Computer Graphics
Computer graphics heavily rely on trigonometric functions to create realistic and visually appealing images. The "sin a+b" formula plays a crucial role in rotating and transforming objects in three-dimensional space. By applying the addition formula, programmers can accurately position and animate objects, resulting in immersive virtual environments and lifelike simulations.
Q&A
1. What is the difference between "sin a+b" and "sin(a + b)"?
The notation "sin a+b" represents the sum of the sines of angles a and b, while "sin(a + b)" represents the sine of the sum of angles a and b. The former is a shorthand notation, whereas the latter is the actual mathematical expression.
2. Can the addition formula be extended to more than two angles?
Yes, the addition formula can be extended to more than two angles. For example, "sin(a + b + c)" can be expressed as "sin a * cos(b + c) + cos a * sin(b + c)." This extension allows us to calculate the sine of the sum of multiple angles.
3. Are there any other trigonometric functions that have similar addition formulas?
Yes, other trigonometric functions, such as cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot), also have addition formulas. These formulas are derived from the fundamental trigonometric identities and are used in various mathematical and scientific applications.
4. Can the addition formula be used to find the difference of two angles?
Yes, the addition formula can be modified to find the difference of two angles. By applying the symmetry property of the sine function, "sin(a – b)" can be expressed as "sin a * cos b – cos a * sin b." This modified formula allows us to calculate the sine of the difference between two angles.
5. How can "sin a+b" be visualized geometrically?
Geometrically, "sin a+b" can be visualized using the unit circle. By representing angles a and b on the unit circle, we can draw lines corresponding to their sines and cosines. The sum of these lines represents the sine of the sum of angles a and b. This visualization aids in understanding the concept and its applications in geometry. | 677.169 | 1 |
can be!
Introduction to Shapes
When we talk about shapes in 1st grade geometry, we often start with the basics. Simple shapes that are easy to identify and draw.
Circles: Imagine a perfectly round pizza or a coin.
Triangles: Think of the pyramids in Egypt or a slice of pie.
Squares: Like a window or a single piece of a checkerboard.
Rectangles: Like a door or a chocolate bar.
These shapes are the building blocks for more complex figures you'll learn about later. Recognizing these shapes in daily life can be a fun game. Try spotting a triangle the next time you eat dinner or finding all the rectangles in your living room.
Understanding Edges and Corners
Now that we have identified some basic shapes, let's understand them a little better. Shapes have edges and corners. For example, a square has 4 straight edges and 4 corners. Similarly, a triangle has 3 edges and 3 corners.
Edges: The straight or curved lines that form the outline of the shape.
Corners: Also known as vertices, they're the point where two edges meet.
You can use your fingers to trace the edges of a shape or count its corners. Play a game with a friend, and see who can identify the edges and corners of different shapes faster!
Solids vs. Flat Shapes
Geometry isn't just about flat shapes; it's also about solid shapes. While a circle is flat, a sphere (like a ball) is solid. A square is flat, but a cube (like a dice) is solid.
Flat Shapes (2D): Have width and height, but no depth. Examples include circles, squares, and triangles.
To understand the difference, try drawing shapes on paper and then finding solid objects around your house that match them.
Combining Shapes
One of the most exciting parts of geometry is combining basic shapes to make new ones. By placing two triangles together, you can form a diamond. Joining many squares side by side can make a rectangle.
A fun activity is to take different colored paper cut-outs of basic shapes and see what new shapes you can create. Perhaps you can design a house using only triangles and rectangles or craft an animal with circles and squares. Let your imagination run wild!
In conclusion, 1st grade geometry is an exciting journey of discovering and understanding the shapes and patterns that make up our world. By recognizing, drawing, and combining shapes, children not only learn the fundamentals of geometry but also develop their spatial and cognitive skills. Dive into the world of shapes with Brighterly and make learning math an exciting adventure!
Basic Geometry Practice Test
Get ready for math lessons with Brighterly! Designed with care, this medium-level test is tailored to challenge and engage young minds, reinforcing their foundational knowledge in geometry.
1 / 20
Which shape has 3 sides and 3 corners?
Square
Circle
Triangle
Rectangle
2 / 20
How many corners does a rectangle have?
2
4
3
0
3 / 20
If you put 2 triangles together, you can make a:
Square
Diamond (Rhombus)
Circle
Pentagon
4 / 20
Which shape is round and has no corners?
Circle
Triangle
Square
Pentagon
5 / 20
Which shape is like a can of soup?
Pyramid
Sphere
Cube
Cylinder
6 / 20
A soccer ball is shaped like a:
Pyramid
Sphere
Cube
Cylinder
7 / 20
How many edges does a square have?
2
3
4
5
8 / 20
Which of these is NOT a flat shape?
Triangle
Rectangle
Circle
Cone
9 / 20
Which shape has 5 sides?
Square
Rectangle
Pentagon
Triangle
10 / 20
A dice is shaped like a:
Sphere
Pyramid
Cube
Cylinder
11 / 20
A pizza slice is shaped most like which of the following?
Triangle
Square
Circle
Pentagon
12 / 20
Which shape rolls but does not slide?
Sphere
Cube
Cylinder
Square
13 / 20
Which shape has all its sides equal in length?
Square
Rectangle
Triangle
Pentagon
14 / 20
How many corners does a pentagon have?
4
3
5
6
15 / 20
A traffic cone is shaped most like:
Cylinder
Cone
Sphere
Cube
16 / 20
If you cut a circle into 4 equal parts, each part is called a:
Triangle
Half
Quarter
Third
17 / 20
Which shape is NOT a solid shape?
Sphere
Cube
Cylinder
Rectangle
18 / 20
How many long sides does a rectangle have?
1
2
3
4
19 / 20
Which of these shapes has the most sides?
Triangle
Square
Pentagon
Rectangle
20 / 20
Which shape looks like a box?
Sphere
Cone
Cube
CylinderAddition Math Practice Test for 3rd Grade – [Hard]
Welcome, young mathematicians and enthusiastic parents! At Brighterly, we believe that every child has the potential to be a math star. 3rd-grade math, especially the domain of basic addition, can be both fun and challenging. And as children progress through their elementary years, understanding addition becomes crucial. This article will walk you through the basics […]
Counting Math Practice Test for 3rd Grade – [Hard]
Hello young mathematicians and curious parents! Today, we're diving deep into the captivating world of basic counting for 3rd graders. Buckle up as we embark on a numerical journey, guided by our trusty educational platform, Brighterly. The Building Blocks: Counting to 1,000 By 3rd grade, many kids are familiar with numbers up to 100. But […]
Money Math Practice Test for 3rd Grade – [Medium]
Hello young explorers and eager learners of the Brighterly community! Have you ever wondered about the coins jingling in your pocket or the colorful bills that grown-ups use to buy things? Let's dive into the exciting world of money in 3rd grade, shall we? Understanding Money Basics: Coins and Bills In every corner | 677.169 | 1 |
Alternate Angles
Alternate angles are
angles
that are on different sides of a
transversal.
A transversal is a line that intersects two other lines.
Alternate interior angles are between the two lines (see figure 1).
Alternate exterior angles are outside the two lines (see figure 2). | 677.169 | 1 |
Tag Archives: 9th class math unit 11 notes
In this post, I am sharing 9th Class Math Unit 11 Notes for the students of the 9th class. Parallelograms and Triangles is the name of 11th chapter and here I am sharing full chapter notes. The good thing is that you can download these Class 9 Math Unit 11 Notes in PDF format. I am sharing these notes with …
I am sharing this post for the students who are looking for Class 9 Mathematics Notes PDf. Here I will Share 9th Class Math Unit 11 Notes in PDF format. Parallelograms & Triangles is the 11th chapter in 9th Textbook and in this post, you will find Unit 11 notes in it. From this post, Students can download 9th Class … | 677.169 | 1 |
Sine Ratio Calculator
Opposite Side Length: Hypotenuse Length:
About Sine Ratio Calculator (Formula)
The Sine Ratio Calculator is a tool used to calculate the sine of an angle in a right triangle based on the lengths of the opposite side and the hypotenuse. It aids in trigonometric calculations and is particularly useful in geometry and engineering applications. The formula for calculating the sine ratio involves considering the lengths of the opposite side and the hypotenuse.
Formula for calculating sine ratio:
Sine Ratio = Opposite Side / Hypotenuse
In this formula, "Opposite Side" represents the side of the right triangle that is opposite to the given angle, and "Hypotenuse" represents the longest side of the right triangle, which is opposite the right angle. Dividing the length of the opposite side by the length of the hypotenuse provides the sine ratio.
For example, let's say a right triangle has an opposite side of length 4 units and a hypotenuse of length 5 units. The sine ratio would be calculated as follows:
Sine Ratio = 4 units / 5 units = 0.8
This means that the sine of the angle, based on the given opposite side and hypotenuse lengths, is 0.8.
The Sine Ratio Calculator simplifies the process of determining the sine of an angle in a right triangle, aiding in trigonometric calculations and engineering applications. By inputting the lengths of the opposite side and the hypotenuse, the calculator quickly provides the sine ratio, allowing individuals and professionals to make accurate measurements and calculations in various fields such as construction, surveying, and physics. | 677.169 | 1 |
1. Coordinate Systems
b. Distance Formula - 2D, 3D and nD
3. Distance Formula in Higher Dimensions (Optional)
The Pythagorean Theorem can be generalized to \(n\) dimensions and then used
to find the distance between two points in \(\mathbb{R}^n\). Consider a
\(n\) dimensional rectangular box with sides \(a_1,a_2,\cdots,a_n\).
In \(\mathbb{R}^n\) the main diagonal of a rectangular box with sides
\(a_1, a_2,\cdots,a_n\) is
\[
d=\sqrt{a_1^2+a_2^2+\cdots+a_n^2}
\]
which is the square root of the sum of the squares of the sides.
Now consider two points \(P=(p_1,p_2,\cdots,p_3)\) and \(Q=(q_1,q_2,\cdots,q_3)\).
The changes in the coordinates are \(|q_1-p_1|,|q_2-p_2|,\cdots,|q_n-p_n|\).
These are the edges of an \(n\) dimensional rectangular box whose main diagonal
is the distance between the points \(P\) and \(Q\).
Equation of an \(n\)-Sphere in \(\mathbb{R}^n\)
By definition, the \(n\)-sphere of radius \(R\)
centered at a point \(C=(a_1,a_2,\cdots,a_n)\) is the set of all
points \(X=(x_1,x_2,\cdots,x_n)\) whose distance from \(C\)
is \(R\). Using the distance formula, the \(n\)-sphere is the set of
all points \(X=(x_1,x_2,\cdots,x_n)\) satisfying the equation
\[
d(C,X)=R
\]
or
\[
\sqrt{(x_1-a_1)^2+(x_2-a_2)^2+\cdots+(x_n-a_n)^2}=R.
\]
The sphere of radius \(R\) centered at a point \(C=(a_1,a_2,\cdots,a_n)\) is
\[
(x_1-a_1)^2+(x_2-a_2)^2+\cdots+(x_n-a_n)^2=R^2.
\]
Find the equation of the \(4\)-sphere for which \(A=(6,1,3,5)\)
and \(B=(2,3,7,-3)\) are endpoints of a diagonal.
Take the coordinates to be \((w,x,y,z)\).
The diameter is the distance from \(A\) to \(B\).
The center is the midpoint of the segment from \(A\) to \(B\). This is found by
averaging the coordinates of \(A\) and \(B\), i.e. \(C=\dfrac{A+B}{2}\).
\((w-4)^2+(x-2)^2+(y-5)^2+(z-1)^2=25\)
The distance from \(A=(6,1,3,5)\) to \(B=(2,3,7,-3)\).
was found in the previous exercise to be \(10\). So the radius is \(5\).
The center is the average of \(A\) to \(B\):
\[
C=\dfrac{A+B}{2}
=\left(\dfrac{6+2}{2},\dfrac{1+3}{2},\dfrac{3+7}{2},\dfrac{5-3}{2}\right)
=(4,2,5,1)
\]
So the equation of the \(4\)-sphere is
\[
(w-4)^2+(x-2)^2+(y-5)^2+(z-1)^2=25
\] | 677.169 | 1 |
Solid Geometry, Volumes 6-9
657. The volume of the frustum of any pyramid is equal to the sum of the volumes of three pyramids whose common altitude is the altitude of the frustum, and whose bases are the lower base, the upper base, and the mean proportional between the bases of the frustum.
Let B and b denote the lower and upper bases, H the altitude, and V the volume of the frustum ABCD-EFGI.
To
prove that V = &H(B+ b + √ B × b).
Proof. Let T-KLM be a triangular pyramid having the same altitude as S-ABCD and its base KLM equivalent to ABCD, and lying in the same plane. Then T-KLM ≈ S-ABCD. § 653 Let the plane EFGI cut T-KLM in NOP.
Hence, if we take away the upper pyramids, we have left the equivalent frustums NOP-KLM and EFGI-ABCD.
But the volume of the frustum NOP-KLM is equal to
H(B+b+√B × b).
. . V = } H (B + b + √ B × b).
§ 656
Q. E. D.
PROPOSITION XXII. THEOREM.
658. The volumes of two triangular pyramids, having a trihedral angle of the one equal to a trihedral angle of the other, are to each other as the products of the three edges of these trihedral angles.
Trim
C
shad
Let V and V' denote the volumes of the two triangular pyramids S-ABC and S'-A'B'C', having the trihedral angles S and S' equal.
V
To prove that
V
SA× SB × SC S'A' × S'B' × S'C'
Proof. Place the pyramid S-ABC upon S'-A'B'C' so that the trihedral S shall coincide with S'.
Draw CD and C'D' L to the plane S'A'B',
and let their plane intersect S'A'B' in S'DD'.
The faces S'AB and S'A'B' may be taken as the bases, and CD, C'D' as the altitudes, of the triangular pyramids C-S'AB and C-S'A'B', respectively.
659. A truncated triangular prism is equivalent to the sum of three pyramids, whose common base is the base of the prism and whose vertices are the three vertices of the inclined section.
Pass the planes AEC and DEC, dividing the truncated prism into the three pyramids E-ABC, E-ACD, and E-CDF.
To prove ABC-DEF equivalent to the sum of the three pyramids, E-ABC, D-ABC, and F-ABC.
Proof. E-ABC has the base ABC and the vertex E.
The pyramid E-ACD≈ B-ACD.
§ 650
For they have the same base ACD, and the same altitude since their vertices E and B are in the line EB to ACD.
But the pyramid B-ACD may be regarded as having the base ABC and the vertex D; that is, as D-ABC.
The pyramid E-CDF ≈ B-ACF.
For their bases CDF and ACF are equivalent,
§ 650
§ 404
since the CDF and ACF have the common base CF and equal altitudes, their vertices lying in the line AD || to CF; and the pyramids have the same altitude, since their vertices E and B are in the line EB || to the plane of their bases.
But the pyramid B-ACF may be regarded as having the base ABC and the vertex F; that is, as F-ABC.
Therefore, the truncated triangular prism ABC-DEF is equivalent to the sum of the three pyramids E-ABC, D-ABC, and F-ABC.
Q.E. D.
660. COR. 1. The volume of a truncated right triangular prism is equal to the product of its base by one third the sum of its lateral edges.
For the lateral edges DA, EB, FC (Fig. 1), being perpendicular to the base ABC, are the altitudes of the three pyramids whose sum is equivalent to the truncated prism.
661. COR. 2. The volume of any truncated triangular prism is equal to the product of its right section by one third the sum of its lateral edges.
For the right section DEF (Fig. 2) divides the truncated prism into two truncated right prisms.
GENERAL THEOREMS OF POLYHEDRONS.
PROPOSITION XXIV. THEOREM. (EULER'S.)
662. In any polyhedron the number of edges increased by two is equal to the number of vertices increased by the number of faces.
in prove whad.
Let E denote the number of edges, V the number of vertices, F the number of faces, of the polyhedron AG.
Proof. Beginning with one face BCGF, we have E
= V.
Annex a second face ABCD, by applying one of its edges to a corresponding edge of the first face, and there is formed a surface of two faces, having one edge BC and two vertices B and C common to the two faces.
Therefore, for 2 faces E = V + 1.
Annex a third face ABFE, adjoining each of the first two faces; this face will have two edges, AB, BF, and three vertices, A, B, F, in common with the surface already formed.
Therefore, for 3 faces
E = V + 2.
In like manner, for 4 faces E
=
V3, and so on.
Therefore, for (F-1) faces E = V + (F− 2).
But F1 is the number of faces of the polyhedron when only one face is lacking, and the addition of this face will not increase the number of edges or vertices. Hence, for F faces | 677.169 | 1 |
Theory of Geometric Series
Lines
Given point (c,d) and slope m, the unique line that satisfies this is the set of all points (x,y) such that
m = y-d/x-c
slope is change in y/change in x
Parallel, Perpendicular
y=2x+4 and y=2x+3 are parallel
y=2x+4 and 2y=4x+8 are same line
y=2x+4 and y = 1/2x+6 meet perpendicularly
Parallel lines never meet, same slope
Perpendicular lines meet once at right angles and slopes are negative reciprocals
Function Variables
Domain = valid inputs to function
Range = what can the function produce
Zeros or Roots = where is f(x)=0
Intersections = Where is f(x) = g(x)
Local maximum is largest value around itself
Local minimum is smallest value around itself
Global is largest overall
Cosine
Tangent
Tangent is the slope of the line with angle theta
Domain is all real numbers except pi/2 + kpi
Range is all real numbers
No max or min. asymptotes at undefined points
Zeros at kpi
Period is pi
Distance Between Points
The distance between two points on the plane is based on the Pythagorean Theorem
|A-B| = sqrt((Xa-Xb)2 +(Ya - Yb)2)
A=(Xa,Ya) B=(Xb,Yb)
Basic Facts
Total human population: 7 billion
USA population: 300 million
Distance from NY to LA: 2500 miles
Distance to the moon: 2.4E5 miles
Distance to the Sun: 1E8 miles
Distance around the equator: 2.5E4 miles
Area of the US: 4E6 square miles
Surface area of the Earth: 2E8 square miles | 677.169 | 1 |
Drawing A Pentagon With a Straightedge And Compass
Whilst we have seen that we can create a pentagon from within the Golden Triangle, it is sometimes convenient to create a pentagon from scratch. Below we will cover how to create a Pentagon using only a Straightedge and Compass. It is my view that Geometry, at least in basic form, should be independent of numbers or any other units. It should focus on proportion and form and is, in effect, scale invariant.
STEP 1: Draw a vertical line.
STEP 2: Draw a circle on the vertical line.
STEP 3: Find the midpoint and draw a line from the centre of the circle to the right.
STEP 4: Find the midpoint of the horizontal line.
STEP 5: Draw a circle with radius equal to quarter the original circle so that its perimeter passes though the centre and is tangential to the outer circle.
STEP 6: Draw a line from the bottom so that it passes through the centre of the inner circle and touches the perimeter on the far side.
STEP 7: Use this length to draw an arc which is tangential to the inner circle at the point marked.
STEP 8: Using your compass, draw an arc with radius equal to the distance from the top of the original circle to where this new arc crosses the original circle.
These are the second and third points of your pentagon.
STEP 9: Taking your compass and placing it on the two points we just created, mark off arcs which are equidistant. If you have been careful in the construction you should be able to create a five petalled flower from these circles as seen in the image attached.
STEP 10: Finally, connect the points we have marked and create your regular pentagon. Marked in bold in the image attached. | 677.169 | 1 |
What is Longitude ? Discuss about the Meridians of Longitude and its types, characteristics and importance
What is Longitude ? Discuss about the Meridians of Longitude and its types, characteristics and importance
Q :- What is Longitude ? Discuss about the Meridians of Longitude and its types, characteristics and importance.
Answer :-
Longitude :-
The angular distance that a straight line drawn from a point on the Earth's surface to the Earth's center makes along the equatorial surface of the earth with the main meridian is called the longitude of that place.
For example the longitude of Calcutta 88°33 east means that Calcutta makes an angle of 88°33 east from the prime meridian along the equatorial plane.
Meridians of Longitude :-
The semi-circular imaginary lines that extend north-south to the equator and the equator of the Earth by connecting equal longitudes on both sides of the main meridian are called meridians.
The English word 'meridian' is derived from the Latin word 'meridias' which means 'middle'. Hence the meridian is called 'Meridian'.
Characteristics of longitude :-
1. Extension :-
The meridian encircles the earth in the North - South direction.
2. Equal Length :-
Every meridian has equal perimeter. Because it starts from North Pole and ends at South Pole.
3. Shape :-
The meridians are imaginary lines of a semicircle and the sum of the angles at the center is 180°.
4. Number and value :-
360 meridians are drawn at 1° intervals. Maximum value of longitude is 180° and minimum value is 0°.
5. Length:-
Meridians are not parallel to each other. Because they meet at two poles. The linear distance between meridians is greatest at the equator. It decreases poleward from the equator and becomes zero at the poles.
6. Centroid:-
The centroid of all meridians is the center of the earth. The radius of the meridians is equal to the radius of the earth. So adding two opposite meridians forms the ellipse.
7. Relation to local time :-
8. Relationship with Axial Lines :-
9. Relationship with climate :-
Differences in climate can be observed between different places located on the same longitude.
10. Angular Sum :-
The angular sum of each longitude is 180°.
The Principal Meridians of Longitude of the Earth are -----
(1) Prime Meridian :-
The imaginary semicircular meridian extending north-south from North Pole to South Pole over London's Greenwich Mean Temple and dividing the earth into eastern and western hemispheres is called Prime Meridian. It makes an angle of 0° at the Earth's center. So its value is 0°. The eastern and western hemispheres are seen together at the prime meridian.
(2) International Date Line:-
Located opposite the equator. Its value is 180° . It is the final boundary between the Eastern and Western Hemispheres. Here too the eastern and western hemispheres can be seen together. It is the meridian that determines times and dates.
Importance of Meridians of Longitude :-
(1) Determination of position :-
How far a place or region of the earth is before the main meridian or what quality and time is determined. The longitude of a place is determined on the basis of the meridian. It is known with the help of Prime Meridian. The Prime Meridian divides the Earth into the Eastern and Western Hemispheres.
(2) Calculating Standard Time :-
The local time of the equator is calculated as the proof time of different countries through the proof time of all the world.
(3) Local time determination :-
Local time is determined based on the position of the midday sun on each meridian.
(4) Boundaries:-
Axis lines define the political boundaries of a country or states within a country. For example, 28° East longitude marks the borders of Libya and Egypt, 120° West longitude marks the borders of the states of California and Nevada.
(5) Date Division:-
The International Date Line is drawn along the 180° meridian which serves to divide dates on Earth. | 677.169 | 1 |
...triangle ABC to the triangle DEF; and the other angles, to which the equal fides are oppofite, fhall be equal, each to each, viz. the angle ABC to the angle DEF, and the angle ACB to DFE. For if the triangle ABC be applied to DEF fo that the point A jnay be on D, and the ftraight line...
...triangle ABC to the triangle DEF; and the other angles, to which the equal fides are oppofite, fhall be equal each to each, viz. the angle ABC •, to the angle DEF, and the » angle ACB to DFE. For, if the triangle ABC be applied to DEF, fo that the point A may be on D, and the ftraight...
...other angles of the cne ABC to the triangle DEF ; and the other angles, to BC which the equal fides are oppofite, fliall be equal, each to each, viz. the angle ABC to the angle DEF, and the angle ACB to DFE. For, if the triangle ABC be applied to the triangle DEF, fo that the point A may be on D, and...
...other angles of the one DEF ; and the other angles, to which the equal fides are oppofite, fhall be equal, each T? to each, viz. the angle ABC to the angle DEF, and the angle ACB to DFE. For if the triangle ABC be applied to DEF fo that the point A may be on D, and the ftraight line...
...to the triangle DEF ; and _ _ the other angles, to BC "BF which the equal fides are oppofite, fhall be equal, each to each, viz. the angle ABC to the angle DEF, and the artgle ACB toDFE. For, if the triangle ABC be applied to the triangle DEF, fo that the point A may...
...equal to the base EF; and the triangle ABC to the triangle DEF; and the other angles to -° v "' J which the equal sides are opposite, shall be equal,...the angle ABC to the angle DEF, and the angle ACB to DFE. For, if the triangle ABC be applied to the triangle DEF, so that the point A may be on D, and...
...equal to one another, their bases shall likewise be equal, and the triangles be equal, and their other angles to which the equal sides are opposite shall be equal, each to each. Which was to be demonstrated. PROP. V. T1IEOR. THE angles at the base of an isosceles triangle are...
...the base BC shall be equal to the base EF; and the triangle ABC to the triangle DEF; and the other angles, to which the equal sides are opposite,! shall be equal each to each,'viz. the angle ABC to theB CE angle DEF, and the angle ACB to DFE. For, if the triangle ABC be... | 677.169 | 1 |
Mitosis Vs Meiosis Worksheet Answers
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Dna The Double Helix Worksheet. For occasion ATGCACATA would code for a special gene than AATTACGGA. The paper is then sent for enhancing to our qualified editors. Any information you enter in your Excel worksheet is saved in a cell. The new cells then receive the directions and data needed to operate. Our world writing... | 677.169 | 1 |
81
Seite 8 ... angle BAC equal to the angle EDF : the AA CE base BC shall be equal to the base EF ; and the tri- angle ABC to the triangle DEF ; and the other angles to which the equal sides are opposite , shall be 8 EUCLID'S ELEMENTS .
Seite 9 ... angle BAC is equal to the angle EDF ; wherefore also the † Hyp . point C shall coincide with the point F , because the straight line AC + is equal to DF : but the point B was + Hyp . proved to coincide with the point E : wherefore the ...
Seite 12 ... angle BDC is equal to the angle BCD ; but BDC has been proved to be greater than the same BCD ; which is impossible ... BAC shall be equal to the angle EDF . For , if the triangle ABC be applied to DEF , so that the point B may be on E ...
Seite 13 ... a given rectilineal angle , that is , to divide it into two equal angles . Let BAC be the given rectilineal angle ; it is re- quired to bisect it . 3. 1 . Take any point D in AB , and from AC cut * off AE equal to AD ; join DE , and | 677.169 | 1 |
Angle Bisector Worksheet Grade 7
Angle Bisector Worksheet Grade 7. Web explore and practice nagwa's free online educational courses and lessons for math and physics across different grades available in english for egypt. Web angle bisectors of triangles worksheet students access live worksheets > english > math > triangles > angle bisectors of triangles angle bisectors of triangles fill in the blanks.
31 Segment And Angle Bisectors Worksheet support worksheet from martindxmguide.blogspot.com
Add to my workbooks (1) download file. Web the angle bisectors worksheet works at ks3 and ks4 and can be completed independently by the learner or as part of your lesson. Worksheets are 5 angle bisectors of triangles, practice work angle bisectors, angle bisectors a, angle.
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Since the angle bisector cuts the angle in half, the. Web this worksheet has 15 angle bisector problems.
Web an angle bisector is defined in a ray that divides a existing angle into two congruent angles. Web straight right grade 7 angles ccss:
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At the top of the worksheet there are 2 worked out examples. Web angle bisector worksheet grade 7.
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All of the problems have multiple steps and are a great review. Md.7 worksheet, fourth grade common core worksheet.
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Web in geometry, the angles are classified as acute, right, obtuse and straight, angle bisectors worksheets will help the students learn about these different types of angles. Web straight right grade 7 angles ccss:
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7.g.b.5 using the diagram shown, ∠ 1 and ∠ 4 are supplementary angles. Some of the worksheets displayed are unit 6 grade 7.
180 ° 20 ° 110 ° 130 °
Web an angle bisector is a line that cuts an angle in half. All of the problems have multiple steps and are a great review. Web the angle bisectors worksheet works at ks3 and ks4 and can be completed independently by the learner or as part of your lesson.
Some Of The Worksheets Displayed Are Unit 6 Grade 7.
Web in geometry, the angles are classified as acute, right, obtuse and straight, angle bisectors worksheets will help the students learn about these different types of angles. You need to measure the angles and find and draw the exact spot where a bisector would be placed. Web explore and practice nagwa's free online educational courses and lessons for math and physics across different grades available in english for egypt.
Learn Read About The Angle Bisector Of A Trilateral And Lever Bisector Theorem.
Web Out Of 100 Ixl's Smartscore Is A Dynamic Measure Of Progress Towards Mastery, Rather Than A Percentage Grade.
Web an angle bisector is defined in a ray that divides a existing angle into two congruent angles. Since the angle bisector cuts the angle in half, the. Worksheets are 5 angle bisectors of triangles, practice work angle bisectors, angle bisectors a, angle. | 677.169 | 1 |
21 Trignometry Formulas for Competitive Exams
One of the most important concepts to understand in the quantitative aptitude section is trigonometry. The study of triangles is basically what trigonometry is, where "trigon" stands for triangle and "metric" for measurement. Trigonometry deals with how a right-angled triangle's angles and sides relate to one another and how to use this information to calculate distances and heights. Read our blog to go through the list of the most important maths trigonometry formulas that a student preparing for competitive exams should not miss.
Firstly the trigonometry table can be used to answer the problems as it is a helpful and easy-to-understand resource. The trigonometry table for angles that are frequently used to solve problems is shown below:
21 Most Important Formulas in Trigonometry
Since many students find it challenging to memorize all of the formulas. It is very important to understand the basic formulas first, as these can be used to derive all other formulas. So are you excited to know what these fundamental identities of trigonometry are? If you're new to trigonometry, or if you just need to revise your concepts, here are the most important maths trigonometry formulas you need to know:
The six basic trigonometric functions are sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot). These functions relate the angles of a right triangle to the ratios of its sides.
How do do class 10 trigonometry?
For class 10 trigonometry, start by understanding the basic trigonometric ratios (sin, cos, tan) and their reciprocal functions. Practice solving problems involving right triangles using these ratios, and gradually move on to more complex concepts like trigonometric identities. | 677.169 | 1 |
The Straight Line: Use Pythagoras' Theorem to Calculate Distance between 2 Points
The Straight Line: Use Pythagoras' Theorem to Calculate Distance between 2 Points
The Straight Line: Use Pythagoras' Theorem to Calculate Distance Between Two Points
Understanding the Concept
Pythagoras' Theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This can be represented as a² = b² + c².
When dealing with coordinates, treat each point as one corner of a right-angled triangle, and the distance between them as the hypotenuse.
The distance d between two points (x₁, y₁) and (x₂, y₂) can therefore be calculated with this formula remembering it comes directly from the theorem itself: d = √[(x₂ - x₁)² + (y₂ - y₁)²].
Applying the Formula
First, determine the coordinates of the two points you are calculating distance between.
Subtract the x-coordinates and subtract the y-coordinates of these two points. This gives you the lengths of the other two sides of your right-angled triangle.
Square these lengths (raise to the power of 2).
Add the squares together.
The square root of this sum gives you the length of the hypotenuse, which is the distance between your two points. | 677.169 | 1 |
Consider the smooth curve in Figure \(7.1.1\). The curvature of the curve at a point is a measure of how drastically the curve bends away from its tangent line, and this curvature is often studied in a multivariable calculus course. The radius of curvature at a point corresponds to the radius of the circle that best approximates the curve at this point. The radius \(r\) of this circle is the reciprocal of the curvature \(k\) of the curve at the point: \(k = \dfrac{1}{r}\text{.}\)
The curvature of a surface (such as the graph of a function \(z = f(x,y)\)) at a particular point is a measure of how drastically the surface bends away from its tangent plane at the point. There are three fundamental types of curvature. A surface has positive curvature at a point if the surface lives entirely on one side of the tangent plane, at least near the point of interest. The surface has negative curvature at a point if it is saddle-shaped, in the sense that the tangent plane cuts through the surface. Between these two cases is the case of zero curvature. In this case the surface has a line along which the surface agrees with the tangent plane. For instance, a cylinder has zero curvature, as suggested in Figure \(7.1.2(c)\).
Figure \(\PageIndex{2}\): The curvature of a surface at a point can be (a) positive; (b) negative; or (c) zero. (Copyright; author via source)
This informal description of curvature makes use of how the surface is embedded in space. One of Gauss' great theorems, one he called his Theorem Egregium, states that the curvature of a surface is an intrinsic property of the surface. The curvature doesn't change if the surface is bent without stretching, and our tireless two-dimensional inhabitant living in the space can determine the curvature by taking measurements.
A two-dimensional bug living in the hyperbolic plane, the projective plane, or the Euclidean plane would notice that a small circle's circumference is related to its radius by the Euclidean formula \(c \approx 2\pi r\text{.}\) In Euclidean geometry this formula applies to all circles, but in the non-Euclidean cases, the observant bug might notice in large circles a significant difference between the actual circumference of a circle and the circumference predicted by \(c = 2\pi r\text{.}\) Large circles about a cup-shaped point with positive curvature will have circumference less than that predicted by Euclidean geometry. This fact explains why a large chunk of orange peel fractures if pressed flat onto a table. Large circles drawn around a saddle-shaped point with negative curvature will have circumference greater than that predicted by the Euclidean formula.
Calculus may be used to precisely capture this deviation between the Euclidean-predicted circumference \(2\pi r\) and the actual circumference \(c\) for circles of radius \(r\) in the different geometries.
Recall that in the hyperbolic plane, \(c = 2\pi \sinh(r)\text{;}\) in the Euclidean plane \(c = 2\pi r\text{;}\) and in the elliptic plane \(c = 2\pi \sin(r)\text{.}\) In Figure \(7.1.3\) we have graphed the ratio \(\dfrac{c}{2\pi r}\) where \(c\) is the circumference of a circle with radius \(r\) in (a) the hyperbolic plane; (b) the Euclidean plane; and (c) the elliptic plane.
In all three cases, the ratio \(\dfrac{c}{2\pi r}\) approaches \(1\) as \(r\) shrinks to \(0\). Furthermore, in all three cases, the derivative of the ratio approaches \(0\) as \(r \to 0^+\text{.}\) But with the second derivative of the ratio we may distinguish these geometries. It can be shown that the curvature at a point is proportional to this second derivative evaluated in the limit as \(r \rightarrow 0^+\text{.}\) We will not derive this formula for curvature but will use this working definition as it appears in Thurston's book [11].
Definition: Curvature of the Space
Suppose a circle of radius \(r\) about a point \(p\) is drawn in a space 1, and its circumference is \(c\text{.}\) The curvature of the space at \(p\) is
the term `space' is intentionally vague here. Our space needs to have a well-defined metric, so that it makes sense to talk about radius and circumference. The space might be the Euclidean plane, the hyperbolic plane or the sphere. Other spaces are discussed in Section 7.5.
Since we are interested in worlds that are homogeneous and isotropic, we will focus our attention on worlds in which the curvature is the same at all points. That is, we investigate surfaces of constant curvature.
Example 7.1.1: The Curvature of a Sphere
Consider the sphere with radius \(s\) in the following diagram, and note the circle centered at the north pole \(N\) having surface radius \(r\text{.}\) The circle is parallel to the plane \(z = 0\text{,}\) has Euclidean radius \(x\text{,}\) and hence circumference \(2\pi x\text{.}\)
But \(x = s\sin(\theta)\) and \(r = \theta\cdot s\text{,}\) from which we deduce \(x = s\sin(\dfrac{r}{s})\text{,}\) and in terms of the surface radius \(r\) of the circle, its circumference is
Note that all the terms of the second derivative after the first have powers of \(r\) in the numerator, so these terms go to \(0\) as \(r \to 0^+\text{,}\) and the curvature of the sphere at the north pole is \(\dfrac{1}{s^2}\text{.}\) In fact because the sphere is homogeneous, the curvature at any point is
\[ k=\dfrac{1}{s^2}\text{.} \]
Example 7.1.2: Curvature of the Hyperbolic Plane
Because hyperbolic geometry is homogeneous and its transformations preserve circles and lengths, the curvature is the same at all points in the hyperbolic plane. We choose to compute the curvature at the origin.
Recall, the circumference of a circle in \((\mathbb{D},\cal{H})\) is \(c = 2\pi \sinh(r).\) To compute the curvature, use the power series expansion
Again, each term of the second derivative after the first has a power of \(r\) in its numerator, so in the limit as \(r \to 0^+\text{,}\) each of these terms vanishes. Thus, the curvature of the hyperbolic plane in (\(\mathbb{D},{\cal H})\) is \(k = -1.\)
Exercises
Exercise \(\PageIndex{1}\)
Use our working definition to show that the curvature of the projective plane in elliptic geometry is \(1\). Recall, \(c = 2\pi\sin(r)\) in this geometry.
Exercise \(\PageIndex{2}\)
Use our working definition to explain why the curvature of the Euclidean plane is \(k = 0\text{.} | 677.169 | 1 |
zamanusa
Instructions:Select the correct answer from each drop-down menu. ∆ABC has vertices at A(11, 6), B(5,...
3 months ago
Q:
Instructions:Select the correct answer from each drop-down menu. ∆ABC has vertices at A(11, 6), B(5, 6), and C(5, 17). ∆XYZ has vertices at X(-10, 5), Y(-12, -2), and Z(-4, 15). ∆MNO has vertices at M(-9, -4), N(-3, -4), and O(-3, -15). ∆JKL has vertices at J(17, -2), K(12, -2), and L(12, 7). ∆PQR has vertices at P(12, 3), Q(12, -2), and R(3, -2). can be shown to be congruent by a sequence of reflections and translations. can be shown to be congruent by a single rotation. | 677.169 | 1 |
29
Page 6 ... joining any two opposite angles of a quadrilateral , Fig . 2 . 36. Plane figures that have more than four sides are generally called polygons , and they receive other particular names according to the number of their sides or angles ...
Page 7 ... joining the extremities of an arc . A segment is any part of a circle bounded by an arc and its chord , Fig . 6 . A semicircle is half the circle , or a segment cut off by a diameter , Fig . 5 ; the half circumference is sometimes ...
Page 9 ... Join AB and B C , and from the point B , with any radius , describe an arc EFG D. From the points A and c , as centres with the same radius , describe arcs cutting the first described arcs at the points EFGD ; through the points of ...
Page 10 ... join E B , and the line thus obtained will be the required perpendicular . Figure 5 . To draw a line making equal angles with two given lines . Let DA , CB , be two converging lines . Draw GH parallel to CB , and IH parallel to D A ...
Page 11 ... join BD , and make B E equal to one third of BD ; make EG perpendicular to BD ; produce the line DC to G , then the points , F and G , with GD and FB for radii , will describe the required ellipsis . Figure 6 . To describe an ellipsis | 677.169 | 1 |
Rope tangent angle over pully given position of offset load
In summary, the conversation discusses the challenge of symbolically resolving the angle of a rope suspending a load between two pulleys. The load is not intended to move horizontally, only vertically. The length of rope let out over the pulleys is a known factor, but the conversation explores the possibility of solving for this angle symbolically rather than using CAD software and excel. Various approaches and equations are suggested and discussed, including one involving the radius of the pulley and the vertical and horizontal distances from the center of the pulley to the attachment point. The conversation also touches on the use of dynamic geometric modeling software to solve this problem.
Jan 29, 2015
#1
Wayland Bugg
7
0Thank you
I don't see why you would assume each pulley has an equal length of line to the rectangle when your two pulleys don't seem to be at the same position vertically.
Jan 29, 2015
#3
drphysica
36
1
Can you be more specific on what you trying to solve because it sounds more complicated than it might actually be. :)
Jan 29, 2015
#4
Wayland Bugg
7
0
LCKurtz said:
I don't see why you would assume each pulley has an equal length of line to the rectangle when your two pulleys don't seem to be at the same position vertically.
You are right they are not at the same height or distance from the load. I misspoke when trying to articulate a visual. But a visual is all intended with that statement. If what I am trying to do were possible, one would only have to enter the vertical and horizontal distances from the center of the pulley to the load attachment point (for example) and could find the angle for any pulley/rope combo.
Jan 29, 2015
#5
Wayland Bugg
7
0
drphysica said:
Can you be more specific on what you trying to solve because it sounds more complicated than it might actually be. :)
I'd like the angle so I can calculate the vertical and horizontal force components for any given position.
If it helps, you could imagine a winch on the other side of the pulley.
one would only have to enter the vertical and horizontal distances from the center of the pulley to the load attachment point (for example) and could find the angle for any pulley/rope combo.
You'd also need to enter the radius R of the pulley.
V = vertical height center of pulley above level attachment point
H = horizontal distance of center of pulley to vertical line through attachment point
You'd also need to enter the radius R of the pulley.
V = vertical height center of pulley above level attachment point
H = horizontal distance of center of pulley to vertical line through attachment point
Thank you for you response! I think that is closer than what I have gotten so far. At least it is another approach. I will work on seeing if this can be adapted to define the angle based only on the amount of line let out at the winch.
I attached another approach I have thought about earlier
Attachments
Capture1.PNG
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Feb 2, 2015
#8
Wayland Bugg
7
0
im not sure this can be solved for in terms of line let out at the winch. is this a problem for dynamic geometric modeling software? does anyone have any experience using such tools that could suggest one with a shorter learning curve?
Is the circle in the upper left of the picture a winch or is it just a pulley?
Which distance in the picture is "line let out of the winch"?
Feb 2, 2015
#10
Wayland Bugg
7
0
Stephen Tashi said:
Is the circle in the upper left of the picture a winch or is it just a pulley?
Which distance in the picture is "line let out of the winch"?
thats just a pulley.
the line let out of the winch isn't directly represented in that picture, but if you could think of the line payed out at the winch, it would pass over the pulley.
in the last image I attached, you see the red line. with the load positioned all the way up, this length is known, so we can say this 'base length' + whatever the winch paid out, is the length of the red line. the blue segment is wrapped around the pulley as the load descends. that's how I divided it up into little chewable understandable pieces, but I could be crazy!
1. What is the formula for calculating the rope tangent angle over a pulley given the position of an offset load?
The formula for calculating the rope tangent angle over a pulley is tanθ = (F × d) / (W - F), where θ is the tangent angle, F is the force of the offset load, d is the distance from the load to the pulley, and W is the weight of the load.
2. How is the rope tangent angle affected by the position of the offset load?
The position of the offset load directly affects the rope tangent angle over the pulley. The closer the load is to the pulley, the larger the rope tangent angle will be. Similarly, the farther away the load is from the pulley, the smaller the rope tangent angle will be.
3. Can the rope tangent angle be negative?
No, the rope tangent angle cannot be negative. It is a measure of the angle formed between the rope and the horizontal line, so it will always be a positive value.
4. How does the weight of the load affect the rope tangent angle?
The weight of the load has a direct impact on the rope tangent angle. As the weight of the load increases, the rope tangent angle also increases. This is because a heavier load exerts a greater force on the rope, causing it to bend at a larger angle over the pulley.
5. Is the length of the rope a factor in calculating the rope tangent angle?
Yes, the length of the rope does play a role in calculating the rope tangent angle. However, it only affects the distance from the load to the pulley (d) in the formula. The longer the rope, the greater the distance from the load to the pulley, resulting in a smaller tangent angle. | 677.169 | 1 |
3-4-5 Rule
If you're looking to set a 90 degree corner but don't have a square, simply use the 3-4-5 rule to locate 90 degrees. All you do is measure any ratio of three, four and five for example if you measure 30cm up one side and 40cm up the other. The diagonal measurement should then be exactly 50cm. If those measurements are correct, then you have found 90 degrees. | 677.169 | 1 |
Distance Formula Worksheet Geometry
Exploring the Distance Formula: A Step-by-Step Guide to Working with a Distance Formula Worksheet
The distance formula is a mathematical tool used to calculate the distance between two points in a given space. While the formula itself is simple to understand, working with it can be complex and daunting for those who are unfamiliar with the concept. However, by following a few simple steps, it is possible to understand and work with the distance formula worksheet successfully.
First, it is important to understand the notation used in the distance formula. The equation is composed of two points, labeled by the letters "A" and "B". These points represent the starting and ending points of the distance being measured. The distance between these two points is then calculated using the formula (A – B)2 + (C – D)2 = E, where "C" and "D" represent the coordinates of the two points and "E" is the distance between them.
Second, it is important to understand the implications of the distance formula. While the equation itself is straightforward, the underlying mathematics can be complex. For example, the equation assumes that the two points are in the same plane, meaning that they are in the same two-dimensional space. If the two points are in different planes, the equation will be different. Therefore, it is important to understand the implications of the equation before working with it.
Third, it is important to understand the implications of the coordinates used in the equation. Coordinates are used to represent the location of the two points in a given space. The coordinates are given in a two-dimensional plane, which is why the equation assumes that the two points are in the same plane. To accurately calculate the distance between two points, the coordinates used must be accurate.
Finally, it is important to know how to use the distance formula worksheet. The worksheet is used to enter the coordinates of the two points and then calculate the distance between them. The worksheet is typically organized with the two points labeled at the top and the coordinates listed below. Once the coordinates are entered, the equation can be solved and the answer will be given.
By following these four steps, anyone can successfully work with the distance formula. However, it is important to keep in mind that the equation assumes that the two points are in the same plane. Therefore, if the two points are in different planes, the equation will be different and the results may not be accurate. Therefore, it is important to be skeptical when working with the distance formula worksheet and to double-check the results before relying on them.
How to Use a Distance Formula Worksheet to Solve Geometry Problems
When attempting to solve geometry problems, many students struggle to identify the best method to use. Although a distance formula worksheet can be a helpful tool for solving geometry problems, it can be difficult to understand how to use it. In this essay, I will argue that although a distance formula worksheet could be useful in solving geometry problems, it may not always be the most efficient method.
To begin with, a distance formula worksheet can be a helpful tool when attempting to solve geometry problems because it provides an organized way to approach the problem. A distance formula worksheet requires students to list all the known variables and then use the appropriate formula to solve the problem. This eliminates the need for students to remember formulas and provides a more organized approach to problem-solving.
On the other hand, using a distance formula worksheet may not always be the most efficient method to solve geometry problems. For example, while the worksheet can provide an organized approach to the problem, it may not be the most efficient way to solve it
In addition, a distance formula worksheet may not always be the best option when attempting to solve geometry problems because it can be difficult to understand how to use it. Since a distance formula worksheet requires students to list out all of the variables and then use the appropriate formula to solve the problem, it can be difficult for students to figure out how to use the worksheet correctly. This can lead to frustration and a lack of understanding of the problem-solving process.
In conclusion, while a distance formula worksheet can be a helpful tool in solving geometry problems, it may not always be the most efficient method Additionally, it can be difficult for students to understand how to use the worksheet correctly, leading to frustration and a lack of understanding of the problem-solving process. Therefore, it is important for students to consider all of the options when attempting to solve geometry problems.
Understanding the Concept of Distance in Geometry with a Distance Formula Worksheet
One of the most fundamental concepts in mathematics is the notion of distance. Distance is a measure of the length between two points in space. It is a foundational concept in geometry, and it is important to understand how this concept works in order to understand more complex topics. Though the concept of distance is simple to explain, it can be difficult to comprehend without the help of a distance formula.
This distance formula worksheet will provide a series of exercises to help students better understand the concept of distance in geometry. The worksheet begins by introducing the concept of distance, then provides a brief explanation of the formula for calculating it. This section is followed by several different equations that can be used to calculate distance. Students will then be asked to solve a few equations using the distance formula.
However, it must be noted that the distance formula is not the only way to calculate distance. In fact, it is only one of many methods for computing the distance between two points. Other methods, such as the Pythagorean theorem, can also be used to calculate distance. However, the distance formula is often the most convenient and straightforward way to calculate distance in geometry.
Though the distance formula worksheet provides a useful introduction to the concept of distance, it is important to keep in mind that this worksheet is not a substitute for understanding the underlying principles of geometry. If students are to truly understand the concept of distance, they must understand how the formula works and the conditions under which it is applicable. Students must also be able to identify the different types of distances and how to compute them using other methods.
In conclusion, the distance formula worksheet can be a helpful tool in helping students understand the concept of distance in geometry. However, it is important to remember that this worksheet is only a starting point. Students must also be able to comprehend the underlying principles of geometry and use other methods in order to accurately calculate distances. It is only with a thorough understanding of the concept of distance that students will be able to apply it in more complex topics.
Conclusion
The Distance Formula Worksheet Geometry is a great way to practice and review the concepts of calculating the distance between two points using the distance formula. It provides students with an opportunity to practice and review the topic in a fun and interactive way. Through this worksheet, students can gain a better understanding of the concept of distance and be better prepared for their next math exam.
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NCERT Class 9 Maths Chapter 10 Circles
Class 9 Maths Chapter 10 Circles
Ex 10.1 Class 9 Maths Question 1. Fill in the blanks. (i) The centre of a circle lies in ___ of the circle. (exterior/interior) (ii) A point, whose distance from the centre of a circle is greater than its radius lies in ____ of the circle, (exterior/interior) (iii) The longest chord of a circle is a ____ of the circle. (iv) An arc is a ____ when its ends are the ends of a diameter. (v) Segment of a circle is the region between an arc and ____ of the circle. (vi) A circle divides the plane, on which it lies, in ____ parts. Solution: (i) interior (ii) exterior (iii) diameter (iv) semicircle (v) the chord (vi) three
Ex 10.1 Class 9 Maths Question 2. Write True or False. Give reason for your answers. (i) Line segment joining the centre to any point on the circle is a , radius of the circle. (ii) A circle has only finite number of equal chords.
(iii) If a circle is divided into three equal arcs, each is a major arc. (iv) A chord of a circle, which is twice as long as its radius, is a diameter of the circle. (v) Sector is the region between the chord and its corresponding arc. (vi) A circle is a plane figure. Solution: (i) True [∵ All points on the circle are equidistant from the centre] (ii) False [ ∵ A circle can have an infinite number of equal chords]
(iii) False [∵ Each part will be less than a semicircle] (iv) True [ ∵ Diameter = 2 x Radius] (v) False [ ∵ The region between the chord and its corresponding arc is a segment] (vi) True [ ∵ A circle is drawn on a plane]
Ex 10.3 Class 9 Maths Question 1. Draw different pairs of circles. How many points does each pair have in common? What is the maximum number of common points? Solution: Let us draw different pairs of circles as shown below:
We have
Figure
Maximum number of common points
(i)
nil
(ii)
one
(«i)
two
Thus, two circles can have at the most two points in common.
Ex 10.3 Class 9 Maths Question 2. Suppose you are given a circle. Give a construction to find its centre. Solution: Steps of construction : Step I : Take any three points on the given circle. Let these points be A, B and C. Step II : Join AB and BC. Step III : Draw the perpendicular bisector, PQ of AB. Step IV: Draw the perpendicular bisector, RS of BC such that it intersects PQ at O.
Thus, 'O' is the required centre of the given drcle.
Ex 10.3 Class 9 Maths Question 3. If two circles intersect at two points, prove that their centres lie on the perpendicular bisector of the common chord. Solution: We have two circles with centres O and O', intersecting at A and B. ∴ AB is the common chord of two circles and OO' is the line segment joining their centres. Let OO' and AB intersect each other at M.
Ex 10.4 Class 9 Maths Question 1. Two circles of radii 5 cm and 3 cm intersect at two points and the distance between their centres is 4 cm. Find the length of the common chord. Solution: We have two intersecting circles with centres at O and O' respectively. Let PQ be the common chord. ∵ In two intersecting circles, the line joining their centres is perpendicular bisector of the common chord.
Ex 10.4 Class 9 Maths Question 3. If two equal chords of a circle intersect within the circle, prove that the line joining the point of intersection to the centre makes equal angles with the chords. Solution: Given: A circle with centre O and equal chords AB and CD are intersecting at E. To Prove : ∠OEA = ∠OED Construction: Draw OM ⊥ AB and ON ⊥ CD. Join OE. Proof: In ∆OME and ∆ONE, OM = ON [Equal chords are equidistant from the centre] OE = OE [Common hypotenuse]
Ex 10.4 Class 9 Maths Question 4. If a line intersects two concentric circles (circles with the same centre) with centre 0 at A, B, C and D, prove that AB = CD (see figure).
Solution: Given : Two circles with the common centre O. A line D intersects the outer circle at A and D and the inner circle at B and C. To Prove : AB = CD. Construction: Draw OM ⊥ l. Proof: For the outer circle, OM ⊥ l [By construction] ∴ AM = MD …(i) [Perpendicular from the centre to the chord bisects the chord]
Ex 10.4 Class 9 Maths Question Let the three girls Reshma, Salma and Mandip be positioned at R, S and M respectively on the circle with centre O and radius 5 m such that RS = SM = 6 m [Given]
Ex 10.4 Class 9 Maths Question Let Ankur, Syed and David are sitting at A, S and D respectively in the circular park with centre O such that AS = SD = DA i. e., ∆ASD is an equilateral triangle. Let the length of each side of the equilateral triangle be 2x. Draw AM ⊥ SD. Since ∆ASD is an equilateral triangle. ∴ AM passes through O. ⇒ SM = 12 SD = 12 (2x) ⇒ SM = x
Ex 10.5 Class 9 Maths Question 1. In figure A,B and C are three points on a circle with centre 0 such that ∠BOC = 30° and ∠ AOB = 60°. If D is a point on the circle other than the arc ABC, find ∠ ADC.
Solution: We have a circle with centre O, such that ∠AOB = 60° and ∠BOC = 30° ∵∠AOB + ∠BOC = ∠AOC ∴ ∠AOC = 60° + 30° = 90° The angle subtended by an arc at the circle is half the angle subtended by it at the centre. ∴ ∠ ADC = 12 (∠AOC) = 12(90°) = 45°
Ex 10.5 Class 9 Maths Question 2. A chord of a circle is equal to the radius of the circle, find the angle subtended by the chord at a point on the minor arc and also at a point on the major arc. Solution: We have a circle having a chord AB equal to radius of the circle. ∴ AO = BO = AB ⇒ ∆AOB is an equilateral triangle. Since, each angle of an equilateral triangle is 60°. ⇒ ∠AOB = 60° Since, the arc ACB makes reflex ∠AOB = 360° – 60° = 300° at the centre of the circle and ∠ACB at a point on the minor arc of the circle.
Ex 10.5 Class 9 Maths Question 7. If diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral, prove that it is a rectangle. Solution: Since AC and BD are diameters. ⇒ AC = BD …(i) [All diameters of a circle are equal] Also, ∠BAD = 90° [Angle formed in a semicircle is 90°] Similarly, ∠ABC = 90°, ∠BCD = 90° and ∠CDA = 90°
Ex 10.5 Class 9 Maths Question 8. If the non – parallel sides of a trapezium are equal, prove that it is cyclic. Solution: We have a trapezium ABCD such that AB ॥ CD and AD = BC. Let us draw BE ॥ AD such that ABED is a parallelogram. ∵ The opposite angles and opposite sides of a parallelogram are equal. ∴ ∠BAD = ∠BED …(i) and AD = BE …(ii) But AD = BC [Given] …(iii)
Ex 10.5 Class 9 Maths Question 10. If circles are drawn taking two sides of a triangle as diameters, prove that the point of intersection of these circles lie on the third side. Solution: We have ∆ABC, and two circles described with diameter as AB and AC respectively. They intersect at a point D, other than A. Let us join A and D.
Ex 10.5 Class 9 Maths Question 11. ABC and ADC are two right angled triangles with common hypotenuse AC. Prove that ∠CAD = ∠CBD. Solution: We have ∆ABC and ∆ADC such that they are having AC as their common hypotenuse and ∠ADC = 90° = ∠ABC ∴ Both the triangles are in semi-circle. Case – I: If both the triangles are in the same semi-circle.
⇒ A, B, C and D are concyclic. Join BD. DC is a chord. ∴ ∠CAD and ∠CBD are formed in the same segment. ⇒ ∠CAD = ∠CBD Case – II : If both the triangles are not in the same semi-circle.
⇒ A,B,C and D are concyclic. Join BD. DC is a chord. ∴ ∠CAD and ∠CBD are formed in the same segment. ⇒ ∠CAD = ∠CBD
Ex 10.5 Class 9 Maths Question 12. Prove that a cyclic parallelogram is a rectangle. Solution: We have a cyclic parallelogram ABCD. Since, ABCD is a cyclic quadrilateral. ∴ Sum of its opposite angles is 180°. ⇒ ∠A + ∠C = 180° …(i) But ∠A = ∠C …(ii) [Opposite angles of a parallelogram are equal] From (i) and (ii), we have ∠A = ∠C = 90°
Similarly, ∠B = ∠D = 90° ⇒ Each angle of the parallelogram ABCD is 90°. Thus, ABCD is a rectangle.
Ex 10.6 Class 9 Maths Question 1. Prove that the line of centres of two intersecting circles subtends equal angles at the two points of intersection. Solution: Given : Two circles with centres O and O' respectively such that they intersect each other at P and Q. To Prove: ∠OPO' = ∠OQO'. Construction : Join OP, O'P, OQ, O'Q and OO'. Proof: In ∆OPO' and ∆OQO', we have
Ex 10.6 Class 9 Maths Question 2. Two chords AB and CD of lengths 5 cm and 11 cm, respectively of a circle are parallel to each other and are on opposite sides of its centre. If the distance between AB and CD is 6 cm, find the radius of the circle. Solution: We have a circle with centre O. AB || CD and the perpendicular distance between AB and CD is 6 cm and AB = 5 cm, CD = 11 cm.
Ex 10.6 Class 9 Maths Question 3. The lengths of two parallel chords of a circle are 6 cm and 8 cm. If the smaller chord is at distance 4 cm from the centre, what is the distance of the other chord from the centre ? Solution: We have a circle with centre O. Parallel chords AB and CD are such that the smaller chord is 4 cm away from the centre.
Ex 10.6 Class 9 Maths Question 4. Let the vertex of an angle ABC be located outside a circle and let the sides of the angle intersect equal chords AD and CE with the circle. Prove that ∠ABC is equal to half the difference of the angles subtended by the chords AC and DE at the centre. Solution: Given : ∠ABC is such that when we produce arms BA and BC, they make two equal chords AD and CE. To prove: ∠ABC = 12 [∠DOE – ∠AOC] Construction: Join AE. Proof: An exterior angle of a triangle is equal to the sum of interior opposite angles. ∴ In ∆BAE, we have ∠DAE = ∠ABC + ∠AEC ……(i) The chord DE subtends ∠DOE at the centre and ∠DAE in the remaining part of the circle.
⇒ ∠ABC = 12 [(Angle subtended by the chord DE at the centre) – (Angle subtended by the chord AC at the centre)] ⇒ ∠ABC = 12 [Difference of the angles subtended by the chords DE and AC at the centre]
Question 1.Answer the following questions.(i) Which are the two main climatic factors responsible for soil formation?(ii) Write any two reasons for land degradation today. (iii) Why is land considered an important resource? (iv) Name any two steps that the government has taken to conserve plants and animals.(v) Suggest three ways to conserve water.Answer.(i) Temperature and | 677.169 | 1 |
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Pagination
To learn the pattern of the side lengths of a 45-45-90 triangle, students complete a gallery walk, a card sort activity starting with using the Pythagorean theorem, and activity to locate if there is an error in a presented problem and if so to identify what the error is. | 677.169 | 1 |
The height of a right-angled triangle drawn from the vertex of the right angle divides the hypotenuse
The height of a right-angled triangle drawn from the vertex of the right angle divides the hypotenuse into segments 2 and 18 find this height
Since CH is the height of triangle ABC, triangles ACH and BCH are rectangular. Let us prove the similarity of triangles ACH and BCH.
Let the value of the angle HAC of the triangle ABC be equal to X0, then the angle ACH = (90 – X) 0.
Angle АСВ = 90, then angle ВСН = (90 – (90 – X) = X0.
The acute angles of the right-angled triangles ACH and BCN are equal, then the triangles are similar in acute angle.
Then in similar triangles AH / CH = CH / BH.
CH2 = AH * BH = 18 * 2 = 36.
CH = 6 cm.
Answer: The length of the CH height is 6 | 677.169 | 1 |
1. Acute Angle
An acute angle is an angle that lies between 0° and 90°. It means an acute angle is greater than 0° but less than 90°. In the above picture, the angle ∠PQR formed by the intersection of PQ and QR which measures 45°. Thus ∠PQR=45° is an acute angle. Common examples of acute angles include 15°, 30°, 45°, 80°, etc.
2. Right Angle
A right angle is an angle whose measure is exactly 90°. It is mainly formed when two straight lines are perpendicular to each other. ∠MNO = 90°.
3. Obtuse Angle
An obtuse angle is an angle that lies between 90° and 180°. It means an obtuse angle is greater than 90° but less than 180°. In the above picture, the angle ∠RST formed by the intersection of RS and ST which measures 110°. Thus ∠RST=110° is an obtuse angle.Common examples of obtuse angles include 100°, 115°, 145°, 160°, etc.
4. Straight Angle
A straight angle is an angle whose measure is exactly 180°. It is one-half of a whole circle. Straight angle is a combination of acute angle and obtuse angle.
5. Reflex Angle
A reflex angle is an angle that lies between 180° and 360º. It means a reflex angle is greater than 180° but less than 360°. [Between straight angle (180º) and a whole turn (360º)]. In the above picture, the angle ∠RST formed by the intersection of RS and ST which measures 220°. Thus ∠RST=220° is an reflex angle. Common examples of obtuse angles include: 190°, 215°, 345°, 350°, etc.
6. Complete Angle
A complete angle is an angle whose measure is exactly 360°. It is one complete revolution equal to 360°. It is also called full rotation or full angle.
Other Types of angles: How to name a pairs of angles
Apart from the types of angles mentioned above, different types of angles are known as pair angles. They include:
Complementary Angle
If the sum of two angles is equal to 90°, then they are called complementary angles. One of the complementary angles is said to be the complement of the other.
In the above picture, ∠PQR + ∠RQS = 55° + 35° = 90°.
Example: 53° To determine the complement, subtract the given angle from 90°.
90° – 53° = 37°, Therefore, the complement of 53° is 37°.
Supplementary Angle
If the sum of two angles is equal to 180°, then they are supplementary angles. One of the supplementary angles is said to be the supplement of the other. In the above picture, ∠ABC + ∠CBD = 60° + 120° = 180°.
Example: 75° To determine the supplement, subtract the given angle from 180°.
180° – 75° = 105°, Therefore, the supplement of 75° is 105°.
Linear Pair Angles
A pair of adjacent angles whose sum is a straight angle, i.e., 180°, is called a linear pair. It means The two angles of a linear pair are always supplementary.
Adjacent Angles
Two angles that share a common side and a common vertex (corner point) are called adjacent angles.
Vertically Opposite Angles
Vertical opposite angles are formed when two lines intersect each other. It is called vertically opposite angles because due to intersection, the angles are opposite to each other. | 677.169 | 1 |
What is a plane in geometry
LearnersCamp has emerged as a highly effective tool in aiding students in their journey to understand and master geometry. By combining interactive elements, such as coefficients in mathematic, personalized learning paths, and a collaborative community, the platform has successfully transformed the way students approach and conquer this challenging mathematical discipline.
Definition and Characteristics:
A plane in geometry is a flat, two-dimensional surface that extends infinitely in all directions. It is defined by a collection of points that lie in the same plane and can be visualized as a perfectly flat surface with no thickness or curvature. Some key characteristics of planes include:
Infinite Extension: A plane extends indefinitely in all directions, encompassing an infinite number of points.
Flatness: Unlike curved surfaces, a plane remains perfectly flat and does not exhibit any curvature.
Uniformity: All points on a plane lie at an equal distance from each other, maintaining a consistent arrangement across the surface.
Equations and Representations:
Planes in geometry can be defined and represented in various ways, including:
Cartesian Equation: A plane can be defined by an equation in the form ax + by + cz + d = 0, where A, B, and C are constants representing the coefficients of the P, N, and Q variables, respectively, and D is a constant representing the distance of the plane from the origin along the direction of the plane's normal vector.
What is a plane in geometry
Vector Equation: A plane can also be described using a vector equation, which specifies a point on the plane and a vector orthogonal to the plane's surface.
Parametric Equations: Parametric equations offer another method of defining a plane, expressing the coordinates of points on the plane in terms of two independent parameters.
Applications in Geometry and Engineering:
The concept of planes finds widespread applications in various fields, including architecture, engineering, computer graphics, and physics. In architecture and engineering, planes are used to represent surfaces such as walls, floors, and ceilings, facilitating the design and construction of structures. In computer graphics, planes serve as the foundation for rendering three-dimensional scenes onto two-dimensional screens, allowing for realistic simulations and visualizations.
Properties and Relationships:
Planes exhibit several important properties and relationships in geometry, including:
Parallelism: Two planes are parallel if they do not intersect and have the same slope or direction.
Intersection: When two planes intersect, they form a line known as the intersection line.
Angle between Planes: The angle between two planes is determine by the angle between their normal vectors.
Distance from a Point to a Plane: The distance from a point to a plane is the perpendicular distance from the point to the plane's surface.
Applications in Geometry and Beyond:
The concept of planes finds widespread applications across various fields, including:
Understanding planes in geometry is essential for various fields, from mathematics and engineering to computer science and beyond. Mastery of plane geometry enables individuals to solve complex problems, visualize spatial relationships, and apply mathematical principles to real-world scenarios.
Conclusion:
Planes are indispensable geometric entities that play a central role in understanding spatial relationships and shapes in geometry and beyond. With their flat, two-dimensional surfaces and infinite extension, planes offer a versatile framework for analyzing geometric concepts, solving problems, and modeling real-world phenomena. Understanding the properties, equations, and applications of planes is essential for mastering geometry and applying geometric principles in various disciplines. | 677.169 | 1 |
what polygon has 5 vertices and 3 sides
Polygon Parts Side - one of the line segments that make up the polygon. These cookies will be stored in your browser only with your consent. There are many more generalizations of polygons defined for different purposes. A regular hexagon, no matter how long its sides, will always have angles that measure \(120^{\circ}\). Draw a polygon with 3 vertices and right angle. The two faces at either end are triangles, and the rest of the faces are rectangular. These cookies ensure basic functionalities and security features of the website, anonymously. A self-intersecting polygon has at least one pair of lines that intersect. 0000151896 00000 n
The segments of a polygonal circuit are called its edges or sides. The figure given below shows a convex and a concave polygon. 0000005535 00000 n
There are many types of polygons. For pentagons, it is only possible to have 3 pentagons meet at a vertex and have the interior angles sum to less than 360: This arrangement corresponds to the dodecahedron. 3. If the polygon is non-self-intersecting (that is, simple), the signed area is, where Out of these, the cookies that are categorized as necessary are stored on your browser as they are essential for the working of basic functionalities of the website. As we know, any polygon has as many vertices as it has sides. This website uses cookies to improve your experience while you navigate through the website. For example, a triangle has three sides, three vertices, and three angles. The polygon is . and Check out 23 similar 2d geometry calculators , How many sides does a polygon have? A square has all its sides equal to 5cm, and all the angles are at 90. ( In a polygon, a vertex is where two edges or sides meet. The names would be 4-gon, or quadrilateral, 44-gon, and 444-gon, respectively. {\displaystyle (x_{j},y_{j}).} hb``a`g`g`` B@Q6oc
. Polygons are classified into various types based on the number of sides and measures of the angles. Also, the vertices of the convex polygon are protruding or pointing outwards. The number of sides of a dodecagon is 12. 4. Convex and concave polygons are types of polygons that differ based on their interior angles. This is called the point in polygon test.[46]. For example, a regular hexagon has six equal sides, and all its interior angles measure to 120 degrees. Polygons are named depending on the number of sides: A regular polygon has sides that are equal in length and equal angles. These are the basic types of regular polygons which we see in our daily life. A simple polygon consists of one boundary. Or, each vertex inside the square mesh connects four edges (lines). The term poly means many and gon means angle. It is also an irregular polygon as the length of all its sides and angles do not measure the same. polyhedra) and is a 3-D solid made out of 2-D polygons. We know that a triangle is composed of 3 vertices and 3 sides,a square is composed of 4 vertices and 4 sides,a pentagon has 5 vertices and 5 sides. These 5 regular polyhedra are sometimes called the Platonic solids, named after the Greek philosopher Plato who believed that the regular polyhedra were the shapes of the 5 classical elements fire (tetrahedron), earth (cube), water (icosahedron), and air (octahedron). The 5 Platonic solids Credit: WikiCommons CC-BY SA 4.0. A convex polygon has all its vertices pointing outwards, whereas a concave polygon has at least one of its vertices pointing inwards. 2023 Science Trends LLC. Also, in a concave polygon, the vertices can be pointing both inwards and outwards. This cookie is set by GDPR Cookie Consent plugin. The measure of the angle may be less than 180 or more than 180. Have a question? {\displaystyle Q_{i,j}} 0000136226 00000 n
The word polygon comes from Late Latin polygnum (a noun), from Greek (polygnon/polugnon), noun use of neuter of (polygnos/polugnos, the masculine adjective), meaning "many-angled". A few examples of a quadrilateral are square, rectangle, rhombus, parallelogram, etc. 1 The triangle, quadrilateral and nonagon are exceptions, although the regular forms trigon, tetragon, and enneagon are sometimes encountered as well. This radius is also termed its apothem and is often represented as a. Let us look more closely at each of those: A vertex (plural: vertices) is a point where two or more line segments meet. An isosceles triangle has two equal sides. The sides of polygons are not limited, and they could have 3 sides, 11 sides, 44 sides, or more. 0000012674 00000 n
A polygonal chain may cross over itself, creating star polygons and other self-intersecting polygons. triangular prism 4 How are the vertices of a polygon determined? > 0000138156 00000 n
6 pi. [9] However, if the polygon is simple and cyclic then the sides do determine the area. In geometry, a polygon is traditionally a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed chain. They can be somewhat tough to imagine. A 0000133584 00000 n
Hexagonal Prism. A pyramid has 4 faces, 8 edges, and 5 vertices. Polygons have, at minimum, three sides and three vertices. The cookie is used to store the user consent for the cookies in the category "Analytics". The area enclosed by a polygon is called thebody. They are made of straight lines, and the shape is "closed" (all the lines connect up). window.__mirage2 = {petok:"AuPV97jSFjEfLfSsb1DVaiyeXMjTSXNpLzzqvA7OoKQ-31536000-0"}; Polygons are among the simplest kinds of shapes and have been rigorously investigated since the time of the ancient Greeks. vertex). Credit: Texample CC-BY SA 2.5. However, a number of polygons are defined based on the number of sides, angles and their properties. 0000156967 00000 n
The term 'Poly' means many and the term 'gon' refers to angle. = 0000103256 00000 n
The cookie is set by GDPR cookie consent to record the user consent for the cookies in the category "Functional". The sides of a polygon are called edges, the point at which two edges of a polygon meet are called vertices, and a, a regular polygon. Draw a polygon with 6 sides. Can banks make loans out of their required reserves? The classification of polygons is made on the basis of the number of angles, their sides, and also according to whether a polygon is regular or irregular. Has two circular bases that are congruent and parallel.5. So this polygon is classified as a concave irregular polygon. Composed of two triangular bases and three . The value of the interior angle of the equilateral triangle always remains 60 and never increases thereby proving it to be a convex polygon. 0000097652 00000 n
Analytical cookies are used to understand how visitors interact with the website. A concave polygon will always have at least one diagonal that falls outside of the area enclosed by the shape. . Other equations, which use parameters such as the circumradius or perimeter, can also be used to determine the area. Therefore, the interior angles of a dodecagon =150 each. {\displaystyle (x_{0},y_{0}),(x_{1},y_{1}),\ldots ,(x_{n-1},y_{n-1})} A polygon with an irregular shape. Example: See the figure of an irregular hexagon, whose vertices are outwards. If the vertices are ordered counterclockwise (that is, according to positive orientation), the signed area is positive; otherwise, it is negative. 0000069266 00000 n
{\displaystyle P=(x_{0},y_{0})} Now let us discuss, types of regular polygon according to the number of sides and measurement of angles. Used as an example in some philosophical discussions, for example in Descartes's. If you have just one its Vertex, if you have more than one that is if you want to make it plural its vertices. Step-by-step explanation: ) Every polygon has an equal number of sides and vertices. 0000136746 00000 n
For example, an equilateral triangle has three equal sides, a square has four equal sides, and a regular hexagon has six equal sides. 0000136270 00000 n
ISSN: 2639-1538 (online). A regular polygon is a polygon that is both equiangular and equilateral. 0000033573 00000 n
0000202923 00000 n
consecutive vertices The endpoints of one side of a polygon. They are made of straight lines, and the shape is "closed" (all the lines connect up). What are vertices and sides of a polygon? 0000156651 00000 n
Polygon comes from Greek. An equilateral pentagon, i.e. [10] Of all n-gons with given side lengths, the one with the largest area is cyclic. By clicking Accept All, you consent to the use of ALL the cookies. It does not store any personal data. Polygons can be classified as convex or concave. Polygons may be characterized by their convexity or type of non-convexity: The property of regularity may be defined in other ways: a polygon is regular if and only if it is both isogonal and isotoxal, or equivalently it is both cyclic and equilateral. 0000142856 00000 n
i holds.[8]. Here is a list of regular polygons from 3 to 10 sides. [citation needed], List of n-gons by Greek numerical prefixes. Put your understanding of this concept to test by answering a few MCQs. They are classified as: A polygon that has all equal sides and angles is called a regular polygon. The different types of the quadrilateral polygon are square, rectangle, rhombus and parallelogram. Is a curved surface of points that are all of the same distance from the center.7. a pentagon whose five sides all have the same length Edges and vertices 5 Internal angle (degrees) 108 (if equiangular, including regular) Which shape has straight sides? Polygons that have more than 20 sides are called n-gons. Beyond decagons (10-sided) and dodecagons (12-sided), mathematicians generally use numerical notation, for example 17-gon and 257-gon.[17]. The following lists the different types of polygons and the number of sides that they have: A triangle is a threesided polygon. 0000015832 00000 n
j 0000100514 00000 n
Most polygons have a special name corresponding to the number of sides (quadrilateral-4 sides, pentagon-5 sides, hexagon, 6 sides, etc) or they can just be called ann-gon, wheren corresponds to the number of sides (3-gon, 4-gon, 5-gon, etc). So the hexagon is the first of the regular polygons that cannot be used to make a regular polyhedron. 4 The cookie is used to store the user consent for the cookies in the category "Other. The cookie is used to store the user consent for the cookies in the category "Other. These trigons or triangles are further classified into different categories, such as: Quadrilateral polygon is also called a four-sided polygon or a quadrangle. For example, a rectangle has all of its angles equal to 90 but all four sides are not equal. 0000003305 00000 n
Our analysis will focus on discovering the area ofregular polygons as those have determinate formulas that are the same regardless of the size of the polygon. For example, a rectangle would turn in to a bowtie-like shape. y This is because a circle is not made out of straight lines, it is a curve. n Polygons are shapes that are made out a finite number of straight lines that close off some area of space. polygon A polygon is a closed 2D shape with straight sides. 0000003198 00000 n
0000242323 00000 n
0 These segments are called its edges or sides, and the points where two of the edges meet are the polygon's vertices (singular: vertex) or corners. "Dergiades, Nikolaos, "An elementary proof of the isoperimetric inequality", "Slaying a geometrical 'Monster': finding the area of a crossed Quadrilateral", 3.0, Not generally recognised as a polygon in the Euclidean plane, although it can exist as a, The simplest polygon which can exist in the Euclidean plane. x A convex polygon has no angles pointing inwards. Our panel of experts willanswer your queries. This cookie is set by GDPR Cookie Consent plugin. Therefore, that arrangement of hexagons can only exist in 2-D space; there is no extra space left for the shape to bend into 3 dimensions. The answer is NO. ( Polygons appear in rock formations, most commonly as the flat facets of crystals, where the angles between the sides depend on the type of mineral from which the crystal is made. 0000034073 00000 n
The simplest polygon such that it is not known if the regular form can be constructed with neusis or not. What are 2 negative effects of using oil on the environment? For any two simple polygons of equal area, the BolyaiGerwien theorem asserts that the first can be cut into polygonal pieces which can be reassembled to form the second polygon. 0000013950 00000 n
So it can be termed as a polygon. 0000207479 00000 n
, [ If a square mesh has n + 1 points (vertices) per side, there are n squared squares in the mesh, or 2n squared triangles since there are two triangles in a square. (straight sides). Some important terms associated with polygons are vertices, edges, and diagonals. These are called triangles. 0000210485 00000 n
3 vertices. 1 Here are a few non-examples of a polygon. Imagine an elastic simple polygon that is picked up and twisted so that some of its sides cross each other. 74 0 obj
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We use cookies on our website to give you the most relevant experience by remembering your preferences and repeat visits. 0 -3 4. The complete question is: What shape has only straight sides more than 3 sides and has fewer than 5 vertices and can be cut into fourths? This page was last edited on 1 May 2023, at 22:26. The pentagon has 5 sides and 5 vertices The second figure has 5sides, all of which areequal and we can observe all of thevertices point outwards, which means all the angles are less than 180. 0000142819 00000 n
y The word "polygon" comes from two Greek words meaning "many angled." All polygons have 3 essential properties: 1. // | 677.169 | 1 |
Elements of Euclid Adapted to Modern Methods in Geometry
From inside the book
Results 1-5 of 43
Page 11 ... sides of a triangle in reference to the opposite angle , and in distinction from the other two sides , is called the base ; the other two sides in reference to the base are often called the legs ; " but the term is scarcely admissible ...
Page 12 ... sides and angles are equal . It is said to be convex when no one of its ... opposite to , or subtending , the right angle is called the hypotenuse . 22 ... sides of a right lined figure is called the perimeter of the figure . The amount ...
Page 19 ... side DF , and the angle A equal to the angle D. It is required to prove that the triangles are equal in every respect ; that is , the bases , the areas , and the other angles , namely , those angles which are opposite to the equal sides ...
Page 22 ... , be equal to the side AC | 677.169 | 1 |
You cannot. An equilateral triangle has 3 lines of symmetry, an
isosceles has one and a scalene none. So there is no triangle with
two lines of symmetry. Of course, you could draw only two of the
three possible lines of symmetry for an equilateral triangle.
Has a triangle got 2 lines of symmetry and 2 lines of rotational symmetry?
First of all, your grammar is terrible. The question should be
"Does a triangle have 2 lines of symmetry and 2 lines of rotational
symmetry? and the answer is no. A triangle can not have 2 lines of
rotational symmetry, because you only rotate the image, you do not
use any lines. | 677.169 | 1 |
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